The Polar Dual of a Convex Polyhedral Set in Hyperbolic Space

Ruth Charney and Michael Davis

Introduction

Let X be a convex polyhedral set in a space of constant curvature. Its “polar dual” P (X) is the set of outward pointing unit normal vectors to the supporting hyperplanes of X. In the spherical and Euclidean cases, P (X) is a subset of the unit sphere. In the hyperbolic case, P (X) is a subset of the unit pseudo-sphere in Minkowski space (sometimes called the “de Sitter sphere”). In all three cases P (X) naturally has the structure of a piecewise spherical cell complex. The spherical cells of P (X) correspond bijectively to the faces of X.

A piecewise spherical cell complex, with its intrinsic metric, is large if there is a unique geodesic between any two points of distance less than π. Equivalently, it is large if it satisfies Gromov’s “CAT(1)-inequality” (cf. [G]).

Piecewise spherical cell complexes play an important role in the study of certain singular metric spaces: the link (or “space of directions”) of a point in such a singular space often has a piecewise spherical structure. The largeness condition is closely related to the notion of nonpositive curvature in the sense of Alexandrov and Gromov. For example, a polyhedron of piecewise constant curvature has curvature bounded from above if and only if the link of each point is large (cf. [G], [B]). A similar result holds for the induced singular metric on the branched cover of a Riemannian manifold (cf. [CD1]).

In his 1986 Princeton Ph.D. thesis [R1], Igor Rivin proved that the polar dual of a convex polytope in hyperbolic 3-space is large. (This was published as the paper [RH] of Rivin and Hodgson.) The proof is a simple geometric argument. (The main result of [RH] is in the converse direction.) The referee has pointed out that some related results, in the smooth category, are proved in [S].

In his 1988 Ohio State Ph.D. thesis [M], Gabor Moussong considered a related situation: he gave a simple, necessary and sufficient condition for a piecewise spherical, simplicial complex with all edge lengths ≥ \frac{π}{2} to be large. The results of both Rivin and Moussong are generalizations of Andreev’s Theorem, [A].

In this paper we use Moussong’s ideas to extend Rivin’s argument to any hyperbolic convex polyhedral set (not necessarily compact) of any dimension. The main result is the following.

Both authors were partially supported by NSF Grants and the Institute for Advanced Study.

Theorem. The polar dual of a convex polyhedral set in hyperbolic space is large.

This theorem is stated, in a sharper form, and proved in Section 4 as Theorem 4.1.1.

A geodesic in P (X) is a special case of a “broken geodesic”, γ = (γ_{1}, …, γ_{k}). This means that each γ_{i} is the arc of a great circle in some cell of P (X) and that the terminal point x_{i} of γ_{i} coincides with the initial point of γ_{i+1}. As in [RH], one associates to each point x_{i} a supporting hyperplane H_{i} of X and to each γ_{i} the codimension two subspace G_{i} = H_{i} ∩ H_{i−1}. (This is explained in subsection 3.1.) Thus, G_{i} and G_{i+1} are hyperplanes in H_{i}. The argument of [RH] then has two steps:

(1) When γ is a geodesic, one shows that G_{i} and G_{i+1} are either parallel or ultraparallel. Hence, there is a region S_{i} in H_{i} bounded by G_{i} and G_{i+1}.

(2) When γ is a closed geodesic the regions S_{i} fit together to give a piecewise isometrically immersed “cylinder” in the ambient hyperbolic space. The exterior dihedral angle along G_{i} is the length of γ_{i}. From this one concludes that the length of γ must be ≥ 2π.

It follows from this last fact that P (X) is large. The proof of (2) generalizes easily to higher dimensions; however, the proof of (1) does not. As we explain in Section 3 (Remark 3.2.3), the argument for (1) in [RH] comes down to the Gauss-Bonnet Theorem in dimension 2 and there is no obvious generalization of this argument to higher dimensions.

Our proof of (1) in higher dimensions occupies all of Section 3. It is based on ideas of [M]. The basic result, Theorem 3.2.2, relates the metric on P (X) to the inner product on the ambient Minkowski space.

In Section 5 we relate the main result to a conjecture of [CD2]. Suppose K^{2m−1} is a piecewise spherical cell complex homeomorphic to the (2m − 1)-sphere. In [CD2] we considered a number κ(K^{2m−1}), defined as a certain alternating sum of normalized volumes of the dual cells to the cells in K^{2m−1}. Hopf’s Conjecture on the sign of the Euler characteristic of a nonpositively curved 2m-manifold leads to the conjecture that (−1)^{m}κ(K^{2m−1}) ≥ 0, whenever K^{2m−1} is large. In subsection 5.2 we recall the following formula of Hopf for the normalized volume v of a 2m-dimensional hyperbolic polytope X^{2m}: (−1)^{m}2v(X^{2m}) = κ(P (X)). (This is a special case of the Gauss-Bonnet Theorem of [AW].) It follows that the sign of κ(K^{2m−1}) is correct when K^{2m−1} is the polar dual of a hyperbolic polytope.

In Section 6, we relate the main result to the study of the space of large piecewise spherical structures on a given simplicial complex.

1. Piecewise spherical polyhedra.

In this section we collect some well-known facts about piecewise spherical polyhedra.

1.1 Convex polyhedral sets. Let V be a finite dimensional real vector space. A subset X of V is a convex polyhedral set if it is defined by a finite number of affine inequalities. In other words, there are linear functions λ_{i} ∈ V^{∗}, and constants c_{i} ∈ ℝ, i = 1, ⋯, k, such that

X = {v ∈ V |λ_{i}(v) ≤ c_{i}, i = 1, ⋯, k}.

A face of X is a (nonempty) subset F of X obtained by changing the inequalities λ_{i}(v) ≤ c_{i} into equalities λ_{i}(v) = c_{i} for i in some subset of {1, ⋯, k}. (Equivalently , F is a face of X if for any two distinct points x, y in X such that (x, y) ∩ F ≠ ∅, we have [x, y] ⊂ F, where (x, y) and [x, y] denote, respectively, the open and closed line segment from x to y.) Let F(X) denote the set of faces of X partially ordered by conclusion and let F_{0}(X) be the set of proper faces of X.

A convex polyhedral set X is called a convex polytope (or a convex cell) if it is compact. Equivalently, X is a convex polytope if it is the convex hull of a finite set.

A polyhedral cone C in V is a convex polyhedral subset defined by a finite number of linear inequalities, i.e.,

C = {v ∈ V |λ_{i}(v) ≤ 0, i = 1, ⋯, k}.

The linear part of C, noted L(C), is the largest linear subspace contained in C. Thus, L(C) is the minimum element in F(C). The cone C is nondegenerate if L(C) = {0}.

For any subset S of V, let Span(S) denote the linear subspace spanned by S. A polyhedral cone C is of full dimension if Span(C) = V

Suppose X is a convex polyhedral set in V and x ∈ X. The tangent cone at x, denoted T_{x}(X), is the polyhedral cone in V (= T_{x}V) consisting of all inward pointing vectors at x, i.e.,

T_{x}(X) = {v ∈ V |x + tv ∈ X for some t > 0}.

If F is a face of X, then this cone is independent of the choice of x in the relative interior of F and is denoted by T (F, X). Explicitly, T (F, X) is the cone determined by the codimension 1 faces of X containing F. The linear part of T (F, X) is given by

L(T (F, X)) = T (F, F) = Span(F − x)

(where F − x is a translate of F to the origin) and

Span (T (F, X)) = Span(X − x).

1.2 Intrinsic metrics. Now identify V with some Euclidean space and let 𝕊(V) denote the unit sphere in V. A spherical convex polyhedral set σ is the intersection of a polyhedral cone C with 𝕊(V). If C is nondegenerate (or equivalently, if σ contains no pairs of antipodal points), then σ is called a spherical convex polytope or a spherical cell. A piecewise spherical cell complex K is a cell complex together with an identification of each cell with a spherical convex polytope. All such cell complexes will be assumed to be locally finite and to have a lower bound on the height of their cells. (The height of a cell σ is the minimum distance between disjoint faces of σ.)

A broken geodesic γ in a K is a sequence (γ_{1}, …, γ_{k}) where each γ_{i} is an oriented geodesic segment (= segment of a great circle) in some cell of K and where the terminal point of γ_{i} is the initial point of γ_{i+1}. The length of γ, denoted by ℓ(γ), is the sum of the lengths of the γ_{i}. The intrinsic metric on K is defined as follows: the distance d(x, y) between points x and y in K is the infimum of the lengths of all broken geodesics from x to y. It follows from the assumptions above that, if x and y belong to the same component of K, then they can be joined by a broken geodesic whose length realizes this infimum (see [Br, Section 1] or [Pa]). Parameterizing by arc length gives a path γ: [a, b] → X such that d(γ(s), γ(t)) = |s − t|, for all s, t ∈ [a, b]. This means that K is a geodesic space (also called a length space). Such an isometric path is a geodesic.

The underlying metric space of a piecewise spherical cell complex with its intrinsic metric is called a piecewise spherical polyhedron.

1.3 Orthogonal joins. Suppose that V and W are disjoint Euclidean spaces, that σ ⊂ 𝕊(V) and τ ⊂ 𝕊(W) are spherical cells, and that C(σ) and C(τ) are the associated polyhedral cones. Then C(σ) × C(τ) is a polyhedral cone in V ⊕ W. The orthogonal join of σ and τ, denoted σ ∗ τ, is the spherical cell defined by

σ ∗ τ = C(σ) × C(τ) ∩ 𝕊(V ⊕ W)

It is obvious how to extend this to a definition of the orthogonal join of two piecewise spherical complexes K_{1} and K_{2}. The resulting intrinsic metric on K_{1} ∗ K_{2} depends only on the intrinsic metrics on K_{1} and K_{2} and not on the particular cell structure.

Suppose K is a piecewise spherical cell complex. The spherical cone on K (or simply the “cone”) is the orthogonal join of K with a point. The k-fold suspension of K is the orthogonal join 𝕊^{k−1} ∗ K. We note that there is no canonical cell structure on 𝕊^{k−1} ∗ K, rather it is partitioned into spherical convex sets of the form 𝕊^{k−1} ∗ σ, where σ is a cell of K or σ = ∅. (For more information about orthogonal joins, see [CD1, Appendix].)

1.4 Links. Suppose that σ is a spherical cell in 𝕊^{n} and that C(= C(σ)) is the associated polyhedral cone in ℝ^{n+1}. Let x be a point in σ and F the face of C which contains x in its relative interior. The tangent cone of σ at x is defined by

T_{x}(σ) = T (F, C) ∩ x^{⊥}.

The link of x in σ is the set of unit vectors in this tangent cone,

Lk(x, σ) = T (F, C) ∩ 𝕊(x^{⊥}).

If τ = F ∩ 𝕊^{n}, then the link of τ in σ is defined by

Lk(τ, σ) = T (F, C) ∩ 𝕊(F^{⊥})

and we have a natural identification

\begin{aligned}Lk(x, σ) & = Lk(τ, σ) ∗ 𝕊(x^{⊥} ∩ F) \\ & = Lk(τ, σ) ∗ 𝕊^{k−1} \end{aligned}

where k = dim τ and x is in the relative interior of τ.

If x is a point in a piecewise spherical complex K, then define

Lk(x, K) = ⋃_{x∈σ} Lk(x, σ)

where the union is taken over all cells σ containing x. Similarly, for a cell τ in K, define

Lk(τ, K) = ⋃_{τ⊂σ} Lk(τ, σ)

where the union is taken over all cells σ properly containing τ. Thus, Lk(τ, K) is naturally a piecewise spherical cell complex. If dim τ = k and x is in the relative interior of τ, then

Lk(x, K) = Lk(τ, K) ∗ 𝕊^{k−1}

so Lk(x, K) is also a piecewise spherical polyhedron.

1.5 Local geodesics. Suppose K is a piecewise spherical cell complex. A closed geodesic in K is an isometric embedding of a circle.

A path γ: [a, b] → K is a local geodesic if for each t ∈ [a, b] there is a neighborhood J of t such that γ|_{J} is geodesic. (A closed local geodesic is defined similarly.)

Suppose γ = (γ_{1}, …, γ_{k}) is a broken geodesic in K, where each γ_{i} is an oriented arc of a great circle from x_{i−1} to x_{i}. If γ is closed (i.e., if x_{0} = x_{k}), then the indices i are interpreted as integers modulo k. At each break point x_{i}, define points γ′_{i} and γ′_{i+1} in Lk(x_{i}, K) as the unit tangent vectors of γ_{i} and γ_{i+1} at x_{i}, oriented outward from x_{i}. Define an “angle” θ_{i} by

(1.5.1)  θ_{i} = d(γ′_{i}, γ′_{i+1})

where d is the intrinsic metric in Lk(x_{i}, K). Then γ is a local geodesic if and only if it satisfies the following “angle condition” (cf. [M, Section 4] or [Br, p. 385]):

(1.5.2)  θ_{i} ≥ π,  i = 0, 1, …, k − 1.

1.5 Large piecewise spherical polyhedra. Any two points in the same component of a piecewise spherical polyhedron K are connected by a geodesic segment. We say that K is large if for any x, y ∈ K with d(x, y) < π, the geodesic segment from x to y is unique. The condition that K is large is equivalent to the condition that it satisfies the “CAT(1)-inequality” of [G, p. 106]. (See [CD1, Theorem 3.1].)

The systole of K, denoted sys(K), is the infimum of the lengths of all closed geodesics in K.

The following lemma is a result of Gromov, [G, §4.2.A and 4.2.B].

Lemma 1.6.1 (Gromov). A piecewise spherical polyhedron K is large if and only if the following two conditions hold:

(i) sys(K) ≥ 2π

(ii) sys(Lk(σ, K)) ≥ 2π for all cells σ in K.

It is fairly obvious that if K is large then (i) and (ii) hold; the proof of the converse uses the angle condition (1.5.2). For the details of the proof, see [B, Theorem 15] and [CD1, Theorem 3.1].

Lemma 1.6.2. Let K be a large piecewise spherical polyhedron. Then any local geodesic of length ≤ π is actually a geodesic.

Proof: The proof uses Alexsandrov’s version of the largeness condition (see [Tr, Théorème 4]). This asserts that, given any geodesic triangle in K of perimeter < 2π, its “angles” must be no greater than those of its comparison triangle in 𝕊^{2}. Let γ = (γ_{1}, …, γ_{k}) be a local geodesic and let x_{i−1} and x_{i} be the endpoints of γ_{i}. We may assume (by subdividing the γ_{i}) that each γ_{i} is a geodesic. The argument proceeds by induction on k.

If k = 1, then γ = (γ_{1}) is geodesic by definition. Let k > 1 and assume by induction that α = (γ_{1}, …, γ_{k−1}) is an actual geodesic from x_{0} to x_{k−1}. Suppose γ is not an actual geodesic. Then the actual geodesic β from x_{0} to x_{k} has length < π. Consider the geodesic triangle (α, γ_{k}, β). The “angle”, between α and γ_{k} at x_{k−1} (in the sense of Alexsandrov), coincides with min{π, θ_{k−1}}, where θ_{k−1} is defined by (1.5.1). By (1.5.2), θ_{k−1} ≥ π. It follows that the comparison triangle in 𝕊^{2} (which exists since perimeter (α, γ_{k}, β) < 2π) must degenerate to a segment. The CAT(1)-inequality then implies that the same holds for (α, γ_{k}, β), that is, γ = β. □

Definition 1.6.3. Suppose K_{1} is a subpolyhedron of a piecewise spherical polyhedron K. Then K_{1} is a locally convex subset of K (or a “totally geodesic subset”) if any local geodesic in K_{1} is actually a local geodesic in K.

The next lemma is an immediate consequence of Lemma 1.6.2.

Lemma 1.6.4. Suppose that K is large and that K_{1} is a locally convex subpolyhedron. Let d_{1} and d denote the intrinsic metrics on K_{1} and K, respectively. Then any local geodesic in K_{1} of length ≤ π is actually a geodesic in K. Thus, if x, y ∈ K_{1} and d_{1}(x, y) ≤ π, then d(x, y) = d_{1}(x, y).

There is also the following simple formulation of local convexity in terms of links. It is an immediate consequence of (1.5.2).

Lemma 1.6.5. Let K_{1} be a subpolyhedron of K. Then K_{1} is locally convex in K if and only if the following condition holds at each point u of K_{1}:

(∗) If x, x′ ∈ Lk(u, K_{1}) and d_{1}(x, x′) ≥ π, then d(x, x′) ≥ π, where d_{1} and d denote the intrinsic metrics in Lk(u, K_{1}) and Lk(u, K), respectively.

2. Polar duality.

In this section we investigate convex polyhedral sets in 𝕊^{n}, 𝔼^{n}, and ℍ^{n}. To such a set X, we will associate a piecewise spherical polyhedron P (X), called the “polar dual” of X. In essence, P (X), consists of the outward pointing unit normal vectors to X. The cell structure on P (X) is induced by the face structure on X: a cell of P (X) corresponds to the outward pointing normals to a face of X.

2.1 Dual Cones. Let V be a finite dimensional real vector space as above. In this subsection, V is equipped with a nondegenerate, symmetric, bilinear form ⟨ , ⟩. If S is a subset of V, then S^{⊥} denotes the linear subspace of V orthogonal to Span(S).

Let C be a polyhedral cone in V. Its dual cone C^{∗} is the polyhedral cone defined by

(2.1.1)  C^{∗} = {w ∈ V |⟨v, w⟩ ≤ 0 for all v ∈ C}.

Lemma 2.1.2 Suppose u_{1}, ⋯, u_{k} ∈ V and

C = {v ∈ V |⟨u_{i}, v⟩ ≤ 0, i = 1, ·s, k}.

Then

(i) C^{∗} = {nonnegative linear combinations of u_{1}, ⋯, u_{k}}
(ii) C^{∗∗} = C
(iii)L(C^{∗}) = C^{⊥}
(iv)Span(C^{∗}) = L(C)^{⊥}

Proof: Let Cū denote the cone of positive linear combinations of u_{1}, ⋯, u_{k}. It is easy to see that Cū ⊂ C^{∗} and hence C^{∗∗} ⊂ C∗ ; ū = C. On the other hand, it is clear from the definition of C^{∗} that C ⊂ C^{∗∗}. This proves (i) and (ii). Part (iii) is clear since w ∈ L(C^{∗}) if and only if w, −w ∈ C^{∗}. Part (iv) follows from (iii) since L(C) = (C^{∗})^{⊥} implies L(C)^{⊥} = Span(C^{∗}). □

In particular, C is of full dimension if and only if C^{∗} is nondegenerate and vice versa. If F is a face of C, then its dual face F̌ is the face of C^{∗} defined by

F̌ = T (F, C)^{∗} = F^{⊥} ∩ C^{∗}

The correspondence F ↦ F̌ is an order-reversing bijection from F(C) to F(C^{∗}). It follows from Lemma 2.1.2 that

Span(F̌) = F^{⊥} ∩ L(C)^{⊥} and L(F̌) = C^{⊥}.

2.2 Polar duals in 𝔼^{n}. Suppose ⟨ , ⟩ is positive definite and identity V with Euclidean space 𝔼^{n}. Let X be a convex polyhedral set in V defined by

X = {v ∈ V | ⟨u_{i}, v⟩ ≤ c_{i}, i = 1, ⋯, k}

and assume without loss of generality that X contains the origin. (The definition of polar dual is translation invariant.) Let W = Span(X). Let C^{∗}(X) denote the polyhedral cone in V spanned by u_{i}, ⋯ u_{k} and W^{⊥} (i.e. C^{∗}(X) is the set of all nonnegative linear combinations of the u_{i} plus a vector in W^{⊥}).

Define the full polar dual of X to be

P̃(X, V) = C^{∗}(X) ∩ 𝕊(V)

and the polar dual of X to be

P (X) = P̃(X) ∩ W = C^{∗}(X) ∩ 𝕊(W)

Since 𝕊(W^{⊥}) ⊂ P̃(X, V), it follows that P̃(X, V) is the orthogonal join

P̃(X, V) = 𝕊(W^{⊥}) ∗ P (X).

Let C(X) denote the dual cone to C^{∗}(X). Then C(X) is the largest polyhedral cone contained in X.

Lemma 2.2.1 Let L_{0} be the linear part of C(X) and let L_{1} be the orthogonal complement of L_{0} in W. Then P (X) is a convex polyhedral set in 𝕊(L_{1}). X is a Euclidean cell (i.e. compact) if and only if L_{1} = W and P (X) = 𝕊(W).

Proof: By Lemma 2.1.1, C^{∗}(X) ⊂ L(C(X))^{⊥} = L_{0}^{⊥}. The first statement of the theorem follows. For the second statement, note that X is compact if and only if C(X) = {0}, or equivalently, C^{∗}(X) = V. □

For each proper face F of X, define

F̌ = T (F, X)^{∗}
σ_{F} = F̌ ∩ P (X)

Note that F̌ ⊂ C^{∗}(X), but since F is not necessarily a face of C(X), F̌ need not be a face of C^{∗}(X). (Figure 1)

Figure 1

Lemma 2.2.2 σ_{F} is a spherical cell. These cells partition P (X) into a piecewise spherical complex.

Proof: It is easy to see that C(X) ⊂ T (F, X) and hence F̌ ⊂ C^{∗}(X). The linear part of F̌ is Span(T (F, X))^{⊥} = Span(X)^{⊥} = W^{⊥}. Thus, F̌ ∩ W is a nondegenerate polyhedral cone, so F̌ ∩ P (X) = F̌ ∩ 𝕊(W) is a spherical cell.

To see that these cells partition P (X), it suffices to observe that the cones T (F, X)^{∗} partition C^{∗}(X), that is, every v ∈ C^{∗}(X) lies in the relative interior of T (F, X)^{∗} for a unique face F of X. Let v ∈ C^{∗}(X) and consider the linear function x → ⟨x, v⟩ on X. The set of maxima of this function is necessarily a face of X which we denote by F_{v}. Then, for an arbitrary face F of X, we have

v ∈ T (F, X)^{∗} ⇔ ⟨w, v⟩ ≤ 0, for all w ∈ T (F, X)
⇔ ⟨x + w, v⟩ ≤ ⟨x, v⟩, for all x ∈ F, w ∈ T (F, X)
⇔ x ∈ F_{v}, for all x ∈ F
⇔ F ⊂ F_{v}

It follows that v lies in the relative interior of a unique T (F, X)^{∗}, namely T (F_{v}, X)^{∗}. □

2.3 Polar duals in 𝕊^{n}. In this subsection, V = ℝ^{n+1}, ⟨ , ⟩ is positive definite, and 𝕊(V) is the unit sphere in V. Let X be a convex polyhedral set in 𝕊(V). Then there is a unique polyhedral cone C = C(X) in V such that X = C ∩ 𝕊(V).

Let C^{∗} be the dual cone.

Points in C^{∗} may be viewed as outward pointing normal vectors to faces of C. The interior points of C^{∗} correspond to normal vectors at the cone point and hence should not be viewed as normals to X. To define the polar dual of X, therefore, we consider ∂C^{∗}. Here, ∂C^{∗} denotes the topological boundary of C^{∗} in V so that if C^{∗} is of dimension less than n + 1, then ∂C^{∗} = C^{∗}. The full polar dual of X is then defined by

P̃(X, V) = ∂C^{∗} ∩ 𝕊(V).

Letting W = Span(C), the polar dual of X is defined by

P (X) = P̃(X, V) ∩ W = ∂C^{∗} ∩ 𝕊(W).

As in the Euclidean case,

P̃(X, V) = 𝕊(W^{⊥}) ∗ P (X).

Lemma 2.3.1 Let L be the linear part of C and let k = dim X = dim W − 1. Then P (X) is the boundary of a spherical cell X^{∗} of dimension k − dim L. In particular, X is a spherical cell (i.e. L = {0}) if and only if P (X) is homeomorphic to a (k − 1)-sphere.

Proof: Let X^{∗} = C^{∗} ∩ 𝕊(W). The cone C^{∗} ∩ W is nondegenerate, so X^{∗} is a spherical cell with ∂X^{∗} = P (X). □

The boundary of a spherical cell is naturally a piecewise spherical complex. The cells of P (X) are the dual faces, σ_{F} = F̌ ∩ P (X) = F^{⊥} ∩ P (X), where F is a proper, nonzero face of C(X).

It is easy to see that these partition P (X) into spherical cells.

2.4 Polar duals in ℍ^{n}. In this subsection, V is (n + 1)-dimensional and the form ⟨ , ⟩ on V is indefinite of type (n, 1). In other words, ⟨ , ⟩ has 1 negative eigenvalue and the rest are positive. (Thus, V could be identified with Minkowski space ℝ^{n,1}.) Let q be the associated quadratic form

q(v) = ⟨v, v⟩ .

We consider the hyperquadrics q^{−1}(−1), q^{−1}(0) and q^{−1}(1) in V. The first, q^{−1}(−1) is a 2-sheeted hyperboloid; choose a component, denote it by ℍ(V) and call it the hyperbolic space of V. The hyperquadric q^{−1}(0) is denoted by 𝕃 and called the light-cone, while q^{−1}(1) is denoted 𝕊_{1}(V) and called the unit pseudo-sphere in V. (𝕊_{1}(V) is sometimes called the “de Sitter sphere.”)

The form ⟨ , ⟩ induces a Riemannian metric on ℍ(V) of constant sectional curvature −1 and a Lorentzian metric on 𝕊_{1}(V) of constant sectional curvature +1. (See [O, pp. 108-114].)

A subspace W of V is spacelike if the restriction of ⟨ , ⟩ to W is positive definite, lightlike if ⟨ , ⟩ restricted to W is positive semidefinite but not definite, and timelike if W contains a vector w with q(w) < 0.

If C is a polyhedral cone in V, then

X = C ∩ ℍ(V)

is called a convex polyhedral subset of ℍ(V); X is called a hyperbolic cell (or a hyperbolic convex polytope) if it is compact. This is the case if and only if C is contained entirely inside the light-cone, i.e., if q(v) < 0 for all nonzero v in C.

Unlike the spherical case, the set X does not uniquely determine the cone C. However, the cone C = C(X) is unique if we require that all codimension 1 faces of C be timelike, i.e.

C = {v ∈ V | ⟨u_{i}, v⟩ ≤ 0, i = 1, ⋯, k}

with q(u_{i}) > 0. (Lightlike and spacelike subspaces do not intersect ℍ^{n} and hence do not affect X.) We will assume this is the case. Even under this hypothesis, however, C may have lower dimensional faces which are not timelike (since, for example, the intersection of two timelike hyperplanes may be a lightlike or spacelike subspace). For the polar dual, we are interested only in outward pointing normals to faces of X or, in other words, to timelike faces of C.

Let C^{∗} be the dual cone to C. The timelike faces F of C correspond to the spacelike faces F̌ of C^{∗}. We therefore let

C_{>0}^{∗} = union of the spacelike faces of C^{∗}.

and define the full polar dual of X to be

P̃(X, V) = C_{>0}^{∗} ∩ 𝕊_{1}(V)

Letting W = Span(C), the polar dual of X is defined as

P (X) = P̃(X, V) ∩ W = C_{>0}^{∗} ∩ 𝕊_{1}(W).

Unlike the spherical and Euclidean cases where P (X) is always homeomorphic to a disk or a sphere, the polar dual of a convex hyperbolic set can have a much more arbitrary topology. However, in the case that X is a hyperbolic cell (i.e. compact), the following lemma states that P (X) is indeed a sphere.

Lemma 2.4.1 Suppose k = dim X = dim C −1. If X is a hyperbolic cell, then C_{>0}^{∗} = ∂C^{∗} and P (X) is homeomorphic to a (k − 1)-sphere

Proof: Let w ∈ ℍ^{n} and let C_{0} = {v ∈ V | q(v) ≤ 0, ⟨v, w⟩ ≤ 0 for all w ∈ ℍ^{n}}. Then C_{0} is the cone bounded by the (positive) light cone and C_{0}^{∗} = C_{0}. (C_{0} is not polyhedral, but the definition (2.1.1) of the dual cone still makes sense.) If X = C ∩ ℍ^{n} is a hyperbolic cell, then C ⊂ int C_{0}; hence, C_{0} = C_{0}^{∗} ⊂ int C^{∗}. It follows that the spacelike faces of C^{∗} are precisely those on its boundary, i.e., C_{>0}^{∗} = ∂C^{∗}.

For w ∈ ℍ^{n}, the restriction of ⟨ , ⟩ to w^{⊥} is positive definite and 𝕊_{1}(V) is homeomorphic to 𝕊(w^{⊥}) × ℝ (≅ 𝕊^{n−1} × ℝ) via the map

𝕊(w^{⊥}) × ℝ ⟶ 𝕊_{1}(V)
(v, r) ⟼ (1 + r^{2})^{\frac{1}{2}} v + rw

Thus, to prove the second statement of the lemma, it suffices to prove that for each v ∈ 𝕊(w^{⊥}), the curve α_{v}(r) = (1 + r^{2})^{\frac{1}{2}} v + rw on 𝕊_{1}(V) intersects ∂C^{∗} precisely once. Restricting to the 2-plane spanned by v and w reduces the problem to the 2-dimensional case, which follows easily from the fact that C_{0} ⊂ int C^{∗}. (Figure 2) □

For any proper face F of C(X), define

P_{F} = F̌ ∩ P (X) = F^{⊥} ∩ P (X).

If F is a timelike face (so F ∩ X is a face of X), then P_{F} turns out to be a spherical cell which we denote by σ_{F} (= P_{F}).

Lemma 2.4.2 If F is timelike, then σ_{F} is a spherical cell. The set {σ_{F} }_{F timelike} partitions P (X) into a piecewise spherical cell complex.

Proof: If F is a timelike face of C then F^{⊥} is a spacelike subspace of V, so F^{⊥} ∩ 𝕊_{1}(V) = 𝕊(F^{⊥}) is a sphere. Also, F̌ ⊂ C_{>0}^{∗} and the linear part of F̌ is W^{⊥}. Thus, σ_{F} = F̌ ∩P (X) = F̌ ∩ W ∩ 𝕊_{1}(V) is a spherical cell in 𝕊(F^{⊥}). Since the faces of C_{>0}^{∗} are precisely the duals of the timelike faces of C, these spherical cells partition P (X) into a piecewise spherical complex. □

Lemma 2.4.3 Suppose F is spacelike. Then F^{⊥} ∩ X is a convex polyhedral set in the hyperbolic space F^{⊥} ∩ ℍ^{n}, and P_{F} = P (F^{⊥} ∩ X). Thus, P_{F} is the polar dual of a convex set of smaller dimension.

Proof: Since F is spacelike, F^{⊥} is timelike. F^{⊥} ∩ C is a polyhedral cone in F^{⊥} whose span is F^{⊥} ∩ W. It is easy to see that the full polar dual of F^{⊥} ∩ X in F^{⊥} is

P̃(F^{⊥} ∩ X, F^{⊥}) = C(X)_{>0}^{∗} ∩ 𝕊_{1}(F^{⊥})

and hence

P (F^{⊥} ∩ X) = C(X)_{>0}^{∗} ∩ 𝕊_{1}(F^{⊥} ∩ W)
= F^{⊥} ∩ P (X).  □

We shall analyze what happens when F is lightlike in the next subsection.

2.5 Cusps. Let 𝕃_{+} denote the positive light cone in ℝ^{n,1}, i.e., 𝕃_{+} = {v ∈ ℝ^{n,1} | q(v) = 0 and ⟨v, x⟩ ≤ 0 for all x ∈ ℍ^{n}}. The sphere at infinity of ℍ^{n}, denoted by S_{∞}, is the projective image of 𝕃_{+} − {0}. Thus, a point in S_{∞} is an open ray in 𝕃_{+} − {0}. Geometrically, a point in S_{∞} can be thought of as the limit point of a geodesic ray in ℍ^{n}.

If y ∈ S_{∞}, then y^{⊥} (a linear subspace of ℝ^{n,1}) is the tangent space along y to 𝕃_{+}. Pick a point x in ℍ^{n}, and consider the affine subspace

A_{y} = x + y^{⊥}.

If v ∈ y, then A_{y} is the locus of those w ∈ ℝ^{n,1} such that ⟨v, w⟩ = constant, where the constant is, of course, ⟨v, x⟩. The horosphere at y (passing through x) is the subset E_{y} of ℍ^{n} defined by

E_{y} = A_{y} ∩ ℍ^{n}.

With the induced metric, E_{y} is isometric to Euclidean (n − 1)-space.

We return to the situation where X is a convex polyhedral set in ℍ^{n}. Let S_{∞}(X) denote the image of (C(X) ∩ 𝕃_{+}) − {0} in S_{∞}. The points in S_{∞}(X) are the points at infinity of X.

For any y ∈ S_{∞}(X), let F_{y} be the unique face of C(X) which contains y in its relative interior. Thus, F_{y} is either timelike or lightlike. The point y is called a cusp point of X if F_{y} is lightlike. If F is any lightlike face of C(X), then there is a unique cusp point y with F_{y} = F; in fact, y = F ∩ F̌.

Suppose y ∈ S_{∞}(X) and that E_{y} is any sufficiently small horosphere at y. (“Sufficiently small” means that E_{y} intersects only those faces of C(X) which contain F_{y}.) Put

Z_{y} = E_{y} ∩ X.

Then Z_{y} is an (n − 1)-dimensional Euclidean convex polyhedral set, well-defined up to similarity.

Lemma 2.5.1. Let y ∈ S_{∞}(X). F_{y} is timelike if and only if Z_{y} is a polyhedral cone. Thus, y is a cusp point if and only if Z_{y} is not a polyhedral cone.

Proof: Let ℰ_{y} be the poset of timelike faces of C(X) which contain y. Then ℰ_{y} indexes the poset of faces of Z_{y}. Thus, Z_{y} is a polyhedral cone if and only if ℰ_{y} contains a minimum (= lower bound). But ℰ_{y} contains such a minimum if and only if F_{y} is timelike (in which case F_{y} is the minimum). □

If y is a cusp point of X, then put

P_{y} = P_{F_{y}} = F_{y}^{⊥} ∩ P (X).

Lemma 2.5.2. If y is a cusp point, then P_{y} can be canonically identified with P (Z_{y}). Thus, the subcomplex P_{y} corresponding to a cusp is isometric to the polar dual of a convex set in 𝔼^{n−1}.

Proof: There is a canonical identification of the horosphere E_{y} with the Euclidean space y^{⊥}/ ⟨y⟩ = 𝔼^{n−1}, well-defined up to translation. (Here, ⟨y⟩ denotes the line determined by y.) On tangent spaces, this identification corresponds to the isomorphism

T_{x}E_{y} = x^{⊥} ∩ y^{⊥}⟶^{≅}y^{⊥}/ ⟨y⟩

induced by the projection y^{⊥} → y^{⊥}/ ⟨y⟩. If F is a timelike face of C(X) containing y, and x ∈ F ∩ E_{y}, then F^{⊥} ⊂ F^{⊥} ∩ x^{⊥} ⊂ y^{⊥} ∩ x^{⊥} = T_{x}E_{y}. It follows that unit normals to F in ℝ^{n,1} can be identified with unit normals to T_{x}(F ∩ E_{y}) in T_{x}E_{y}, and hence with unit normals to F ∩ E_{y} in E_{y}, viewed as 𝔼^{n−1}. □

2.6 Links in P (X). Let us summarize the results of sections 2.3-2.5. Let X be a convex polyhedral set in 𝔼^{n}, 𝕊^{n} or ℍ^{n} and let C = C(X) be the cone determined by X (as in 2.3-2.5). If Y is a face of X, then F_{Y} = C(Y) is a face of C. (In the spherical and hyperbolic cases, F_{Y} is the unique face such that Y = X ∩ F_{Y}.) Associated to Y is a spherical cell σ_{Y} = Y̌ ∩ P (X). (In the spherical and hyperbolic cases, Y̌ ≡ F̌_{Y} , σ_{Y} ≡ σ_{F_{Y}}; in the Euclidean case, Y̌ ⊂ F̌_{Y}, but they need not be equal.) Let F_{0}(X) denote the set of proper faces of X ordered by inclusion. Let F_{0}(X)^{op} denote this set with the opposite ordering.

Proposition 2.6.1 Let X be a convex polyhedral set in 𝔼^{n}, 𝕊^{n}, or ℍ^{n}. Then the polar dual of X, P (X) is a piecewise spherical cell complex whose poset of cells {σ_{Y} } is naturally identified with F_{0}(X)^{op}. If X is a cell of dimension k in 𝔼^{n}, 𝕊^{n}, or ℍ^{n}, then P (X) is homeomorphic to a (k − 1)-sphere.

Lemma 2.6.2 Let X be a convex polyhedral set in 𝔼^{n}, 𝕊^{n}, or ℍ^{n} and let W = Span(C(X)). Let Y ∈ F_{0}(X) and let y be a point in the relative interior of σ_{Y}. Then

(1) Lk(y, P (X)) = P̃(Y, y^{⊥} ∩ W)
(2) Lk(σ_{Y} , P (X)) = P (Y)

Proof: The correspondence F ↦ F̌ gives a bijection between faces of C and faces of C^{∗} such that

F = ˇ F̌ = T (F̌, C^{∗})^{∗}.

Thus, F^{∗} = T (F̌, C^{∗}) (where F^{∗} is the dual cone in V to F, viewed as a polyhedral cone). Let F = F_{Y} = C(Y). Then y lies in the relative interior of F̌, so

F^{∗} = T_{y}(C^{∗}).

The faces of F^{∗} are of the form T (F̌, Ǧ) where G is a face of F. In the hyperbolic case, T (F̌, Ǧ) is spacelike if and only if Ǧ is spacelike (since Span(T (F̌, Ǧ)) = Span(Ǧ)), so

F_{>0}^{∗} = T_{y}(C_{>0}^{∗}).

Thus, in the spherical and Euclidean cases,

Lk(y, P̃(X)) = 𝕊(y^{⊥}) ∩ T_{y}(C^{∗})
= 𝕊(y^{⊥}) ∩ F^{∗}
= P̃(Y, y^{⊥}).

The same holds in the hyperbolic case if we replace C^{∗} by C_{>0}^{∗} and F^{∗} by F_{>0}^{∗}. Intersecting both sides of the equation with W gives part (1) of the lemma; intersecting with Span(Y) (= (Y̌)^{⊥}) gives part (2). □

3. The intrinsic metric on P (X).

3.1 A geometric interpretation of local geodesics in P (X). There is an interesting interpretation of a local geodesic in P (X) in terms of the hyperbolic geometry of X. This relationship is explained in [RH]. We shall now recall it.

Each point v in 𝕊_{1}^{n} corresponds to an oriented hyperbolic hyperplane

H_{v} = v^{⊥} ∩ ℍ^{n}.

If v ∈ P (X), H_{v} is called a supporting hyperplane of X. This terminology is appropriate since v lies in P (X) if and only if X lies in the half-space

D_{v} = {u ∈ ℍ^{n} | ⟨u, v⟩ ≤ 0}

and X ∩H_{v} is nonempty. (If v belongs to the relative interior of the spherical cell σ_{Y}, then H_{v} ∩ X = Y.)

Suppose that σ_{Y} is a cell of P (X), that v_{0} and v_{1} are points in distinct faces of ∂σ_{Y} and that γ is the geodesic segment in σ_{Y} from v_{0} to v_{1}. Put

G = γ^{⊥} ∩ ℍ^{n} = H_{v_{0}} ∩ H_{v_{1}}.

Then G is a totally geodesic codimension two subspace of ℍ^{n}. The length ℓ(γ) of γ is cos^{−1} ⟨v_{0}, v_{1}⟩. Hence, ℓ(γ) measures the exterior dihedral angle between H_{v_{0}} and H_{v_{1}}. Corresponding to the points of γ, there is a 1-parameter family of supporting hyperplanes obtained by rotating H_{v_{0}} to H_{v_{1}} about G through an angle of ℓ(γ).

Next suppose that γ = (γ_{1}, …, γ_{k}), is a broken geodesic in P (X), where γ_{i} is a geodesic segment in some cell from v_{i−1} to v_{i}, as in the previous paragraph. Put H_{i} = H_{v_{i}} and G_{i} = (γ_{i})^{⊥} ∩ ℍ^{n} = H_{i−1} ∩ H_{i}. Thus, γ gives us a 1-parameter family of supporting hyperplanes “rolling” along the boundary of the convex body X and containing H_{0}, H_{1}, …, H_{k−1} in succession.

Consider the (n−1)-dimensional hyperbolic space H_{i}. Then G_{i} and G_{i+1} are oriented hyperplanes in H_{i} with corresponding unit normal vectors γ′_{i} and γ′_{i+1} in Lk(v_{i}, P (X)). Suppose that the broken geodesic γ is actually a local geodesic, i.e., d(γ′_{i}, γ′_{i+1}) ≥ π, (cf. (1.5.2)). We shall prove in Proposition 3.3.3 below that this implies that the hyperplanes G_{i} and G_{i+1} do not intersect in H_{i}. Hence, G_{i} and G_{i+1} bound an infinite (n−1)-dimensional “strip” S_{i} in H_{i}.

Finally, we suppose that γ = (γ_{1}, …, γ_{k}) is a closed local geodesic. Then we get a sequence of strips (S_{0}, …, S_{k−1}) such that S_{i} ⊂ H_{i} and S_{i−1} ∩ S_{i} = G_{i}. Let S_{γ} = S_{0}∪ ⋯ ∪S_{k−1} be the union of these strips in ℍ^{n}. We note that, even when the indices i and j are not consecutive (modulo k), the strips S_{i} and S_{j} might have nonempty intersection. To eliminate these extraneous intersections, one introduces an abstract (n−1)- dimensional hyperbolic manifold Ŝ_{γ} formed by taking the quotient of the disjoint union ∐ S_{i} by the equivalence relation which identifies the boundary components of S_{i−1} and S_{i} which correspond to G_{i}. The natural map Ŝ_{γ} → S_{γ} ⊂ ℍ^{n} is then a piecewise totally geodesic immersion.

The (n − 1)-manifold Ŝ_{γ} is a “hyperbolic cylinder.” That is to say, it is isometric to ℍ^{n−1}/ ⟨ρ_{γ}⟩, where ρ_{γ} is either a parabolic or hyperbolic isometry of ℍ^{n−1} and where ⟨ρ_{γ}⟩ denotes the infinite cyclic group generated by ρ_{γ}. Thus, to each closed local geodesic γ in P (X) we have associated an immersed hyperbolic cylinder S_{γ} in ℍ^{n} such that the length of γ is the sum of the exterior dihedral angles in S_{γ}. This cylinder will be used in Section 4 to prove the main theorem (Theorem 4.1.1).

3.2 A version of a result of Moussong.

Our goal in this subsection is to prove Theorem 3.2.2, below, which relates the intrinsic metric on P (X) to the Lorentzian inner product on ℝ^{n,1}. We begin with a preliminary lemma which clarifies the statement of Theorem 3.2.2.

Lemma 3.2.1. If v, w ∈ P (X), then ⟨v, w⟩ ≤ 1, with equality if and only if v = w.

Proof: Suppose that v, w are distinct vectors in P (X) with ⟨v, w⟩ ≥ 1. Then the supporting hyperplanes H_{v} and H_{w} do not intersect in ℍ^{n}. Thus, for any x in H_{v}, the quantity ⟨w, x⟩ is never zero. If it is negative, then the half-space D_{v} is contained in D_{w}; if positive, then D_{w} ⊂ D_{v}. In either case, H_{v} and H_{w} cannot both be supporting hyperplanes, a contradiction. □

Theorem 3.2.2. Suppose v, w are points in P (X) with ⟨v, w⟩ > −1. Then

d(v, w) ≤ cos^{−1} ⟨v, w⟩ ,

where d is the intrinsic metric on P (X). In fact, suppose neither of the following conditions hold:

(i) v and w belong to some cell of P (X),

(ii) v and w belong to P_{y} for some cusp point y of X.

Then d(v, w) < cos^{−1} ⟨v, w⟩.

The above theorem is a modification of Lemma 9.7 in the thesis of G. Moussong [M]. Although Moussong was working in a slightly different context, both the statement and proof of Theorem 3.2.2 are essentially the same as in [M] (our case is somewhat easier).

Remark 3.2.3. When X is a 2-dimensional convex polyhedral set in ℍ^{2} there is an easy geometric proof of Theorem 3.2.2, along the lines of the proof of Lemma 3.4 in [RH], which we now recall. Suppose v, w ∈ P (X) and ⟨v, w⟩ ≥ − 1. Then the supporting lines H_{v} and H_{w} intersect at a point q_{0} which either lies in ℍ^{2} or on the circle at infinity (when ⟨v, w⟩ = −1). Let Q be the polygon, exterior to X, which is bounded by H_{v}, H_{w} and some of the edges of X as indicated in Figure 3, below. Let q_{1}, …, q_{k} be the other vertices of Q and let α, θ_{1}, …, θ_{k} be the indicated angles. Then α = cos^{−1} ⟨v, w⟩ and d(v, w) = θ_{1} + ⋯ + θ_{k}.

For any finite area polygon Q in ℍ^{2} (convex or not), the Gauss-Bonnet Theorem states that the sum of the exterior angles of Q is equal to 2π + Area(Q). In our case, the exterior angle at q_{0} is α, at q_{1} it is π − θ_{1} and at q_{k} it is π − θ_{k}, while, for 2 ≤ i ≤ k − 1 the exterior angle at q_{i} is −θ_{i} (a negative number). So,

α + (π − θ_{1}) − (θ_{2} + ⋯ + θ_{k−1}) + (π − θ_{k}) = 2π + Area(Q)

Hence, α = ∑ θ_{i} + Area(Q), that is,

cos^{−1} ⟨v, w⟩ = d(v, w) + Area(Q).

Thus, d(v, w) ≤ cos^{−1} ⟨v, w⟩ and the inequality is strict unless Area(Q) = 0. If Area(Q) = 0, then H_{v} and H_{w} are actually supported by edges of X, and q_{0} is a vertex of X (possibly at ∞) and so we are in situation (i) or (ii) of the theorem.

Before giving the details of the proof of Theorem 3.2.2, let us sketch the rough idea. Suppose v and w are distinct points in P (X). By Lemma 3.2.1 we always have ⟨v, w⟩ < 1. Hence, the condition that ⟨v, w⟩ > −1 means that |⟨v, w⟩| < 1, i.e., that v and w span a spacelike subspace of ℝ^{n,1}. The cone generated by v and w then intersects 𝕊_{1}^{n} in a circular arc δ of length cos^{−1} ⟨v, w⟩. In general, δ will not lie in P (X). Let u denote the unit tangent vector to δ at v. In a sequence of lemmas, stated below (in particular, in Lemma 3.2.8), we show that one can always find a vector z ∈ Lk(v, P (X)) so that as x moves along the geodesic γ in direction z, the quantity cos^{−1} ⟨x, w⟩ decreases at least as fast as it does when x moves along δ. (The condition that z have this property is just that ⟨u, z⟩ ≥ 1.) Using a compactness argument, we show that we can extend γ to a path from v to w of length at most that of δ.

Suppose G is a subspace of ℍ^{n} and y ∈ S_{∞}. We say that G meets y if there is a geodesic ray in G which limits at y.

Lemma 3.2.4. Suppose that v, w ∈ P (X) are such that H_{v} ∩ H_{w} ∩ X = ∅. If y is a point in S_{∞}(X) which meets H_{v} and H_{w}, then y is a cusp point of X.

Proof: Let E_{y} be a small horosphere at y and Z_{y} = E_{y} ∩ X. Then Z_{y} is a Euclidean convex polyhedral set and v^{⊥} ∩ E_{y} and w^{⊥} ∩ E_{y} are supporting hyperplanes of Z_{y} which do not intersect. It follows that Z_{y} is not a polyhedral cone (if it were, the intersection of any two supporting hyperplanes would contain the minimum face). By Lemma 2.5.1, y is a cusp point.

Lemma 3.2.5. Suppose that u is a unit vector in C^{∗}(X). Let d(H_{u}, X) denote the distance (in ℍ^{n}) from H_{u} to X.

(i) If d(H_{u}, X) > 0, then there is a unique point x_{0} in X such that d(H_{u}, X) = d(H_{u}, x_{0}).

(ii) If d(H_{u}, X) = 0, then either (a) H_{u} is a supporting hyperplane for X (i.e., u ∈ P (X)) or

(b) H_{u} meets S_{∞}(X) in a single point.

(iii) There is a vector z in P (X) such that ⟨u, z⟩ ≥ 1. Moreover, if d(H_{u}, X) > 0, then z can be chosen so that ⟨u, z⟩ > 1.

Proof: (i) If α(t) is any infinite geodesic ray in X, then

lim_{t→∞} d(H_{u}, α(t)) = \{ \begin{matrix}0; & if α(t) and H_{u} meet at infinity \\ ∞; & otherwise \end{matrix}

Since X is geodesically convex and since the restriction of the function x → d(H_{u}, x) to any geodesic segment is strictly convex, it follows that if d(H_{u}, x) is bounded away from 0, then there is a unique point x_{0} in X where the minimum value is obtained.

(ii) Suppose α_{1}(t) and α_{2}(t) are two geodesic rays in X which both meet H_{u} at infinity. Let y_{1} and y_{2} be the corresponding limit points in S_{∞}(X). If y_{1} ≠ y_{2} then, by convexity, the infinite geodesic from y_{1} to y_{2} is contained in H_{u} ∩ X and hence, H_{u} is a supporting hyperplane for X. Therefore, if H_{u} is not a supporting hyperplane, then y_{1} = y_{2}, which proves (ii).

(iii) First, suppose that d(H_{u}, X) > 0. Then we choose z to be the parallel transport of u to the closest point x_{0}. It is then geometrically clear that H_{z} is a supporting hyperplane. This can also be checked by the following algebraic calculation. The vector z is the orthogonal projection of u onto (x_{0})^{⊥}, normalized to have unit length, i.e.,

z = \frac{u + ⟨u, x_{0}⟩ x_{0}}{(1 + ⟨u, x_{0}⟩^{2})^{\frac{1}{2}}}

Since d(H_{u}, x) = − sinh ⟨u, x⟩ (see [T, Section 2]), the function x → ⟨u, x⟩ takes its maximum value on X at x_{0}, i.e.,

⟨u, x_{0}⟩ ≥ ⟨u, x⟩ , for all x ∈ X.

Since the inner product of any two points in ℍ^{n} is ≤ − 1,

⟨x_{0}, x⟩ ≤ − 1

Hence, for any x ∈ X,

⟨u + ⟨u, x_{0}⟩ x_{0}, x⟩ = ⟨u, x⟩ + ⟨u, x_{0}⟩ ⟨x_{0}, x⟩ ≤ ⟨u, x⟩ − ⟨u, x⟩ = 0.

So, u + ⟨u, x_{0}⟩ x_{0} belongs to C^{∗}(X) and therefore, so does its positive multiple z. Since ⟨z, x_{0}⟩ = 0 and ⟨z, z⟩ = 1, we see that z ∈ P (X). Moreover,

⟨u, z⟩ = (1 + ⟨u, x_{0}⟩^{2})^{\frac{1}{2}} > 1,

as claimed.

Finally, suppose that d(H_{u}, X) = 0. By (ii) there are two cases to consider. If H_{u} is a supporting hyperplane, then put z = u. Otherwise, let y be the unique point in S_{∞}(X) which meets H_{u}. Let E_{y} be a sufficiently small horosphere at y and Z_{y} = E_{y} ∩ X. Then Z_{y} is a Euclidean convex polyhedral set and u^{⊥} ∩ E_{y} is an affine hyperplane which bounds a halfspace containing Z_{y}. let H′ be the oriented affine hyperplane in E_{y} obtained by translating u^{⊥} ∩ E_{y} to a closest point in Z_{y}. Then there is a unique z ∈ 𝕊_{1}^{n} such H_{z} and H_{u} are parallel and meet at infinity at y and such that H_{z} ∩ E_{y} = H′. Clearly, z ∈ P (X) and ⟨u, z⟩ = 1, as desired.

In the next two lemmas and in subsection 3.3 we must deal with the case where the convex polyhedral set might be of smaller dimension than the ambient hyperbolic space. In other words, the full polar dual might not coincide with the polar dual. We set up some notation to deal with this situation below.

Notation 3.2.6. Suppose Y is a convex polyhedral set in ℍ^{n−1} (possibly of dimension < n − 1). Let u ∈ ℝ^{n−1,1}. If u is spacelike, then ū = q(u)^{− \frac{1}{2}} u denotes its normalization to a unit vector. Denote by u_{1} and u_{2} the orthogonal projections of u onto Y^{⊥} and Span(Y), respectively. Identify Y^{⊥} with ℝ^{ℓ} and Span(Y) with ℝ^{n−ℓ−1,1}. If u_{2} is any spacelike vector in ℝ^{n−ℓ−1,1}, then G_{u_{2}} denotes the hyperbolic hyperplane (u_{2})^{⊥} ∩ ℍ^{n−ℓ−1}. As in 2.5, P̃(Y) = 𝕊^{ℓ−1} ∗ P (Y) (where the ambient space V = ℝ^{n−1,1} is omitted from the notation).

Lemma 3.2.7. Let u be a unit vector in ℝ^{n−1,1} such that u_{2} ∈ C^{∗}(Y). Then there is a vector z ∈ P̃(Y) such that ⟨u, z⟩ ≥ 1. Moreover, z can be chosen so that the inequality is strict except possibly in the following three cases:

(a) u_{2} = 0,

(b) u_{2} is spacelike and d(G_{u_{2}}, Y) = 0,

(c) u_{2} is lightlike and u_{2} ∈ S_{∞}(Y).

Proof: There are four cases to consider as the vector u_{2} is zero, timelike, spacelike or lightlike.

Case 1: u_{2} = 0. Put z = u. Then z lies in the suspension sphere and hence, in P̃(Y). Furthermore, ⟨u, z⟩ = 1.

Case 2: u_{2} is timelike. Then q(u_{1}) = 1 − q(u_{2}) > 1. In particular, u_{1} ≠ 0. Put z = ū_{1}. Again, z lies in the suspension sphere and ⟨u, z⟩ = q(u_{1})^{\frac{1}{2}} > 1.

Case 3: u_{2} is spacelike. Apply Lemma 3.2.5 to ū_{2} and Y to find z_{2} ∈ P (Y) so that ⟨ū_{2}, z_{2}⟩ ≥ 1. Put z = u_{1} + q(u_{2})^{\frac{1}{2}} z_{2}. Then q(z) = q(u_{1}) + q(u_{2}) = 1, so z ∈ 𝕊^{ℓ−1} ∗ P (Y). Furthermore, ⟨u, z⟩ = q(u_{1})+q(u_{2}) ⟨ū_{2}, z_{2}⟩ ≥ q(u_{1})+q(u_{2}) = 1, and the inequality is strict if and only if ⟨ū_{2}, z_{2}⟩ > 1. By Lemma 3.2.5 (iii), we can choose z_{2} so that ⟨ū_{2}, z_{2}⟩ > 1 unless d(G_{u_{2}}, Y) = 0.

Case 4: u_{2} is lightlike. In this case one might try the approach in Case 2 or the one in Case 3. Both methods work; however, only the second method yields a strict inequality. Since q(u_{2}) = 0, q(u_{1}) = 1 − q(u_{2}) = 1. The first method is to choose z = u_{1}. Then z is in the suspension sphere and ⟨u, z⟩ = 1. Let y be the point in S_{∞} determined by u_{2}. The second method can be applied whenever y ∉ S_{∞}(Y). As in Lemma 3.2.5, we find a point x_{0} in Y which is closest to y. Explicitly, consider the smallest horosphere at y which intersects Y. The intersection must be a single point x_{0}. Let G be the hyperbolic hyperplane in ℍ^{n−ℓ−1} which is tangent to the horosphere at x_{0} and let z_{2} be the unit normal to G which points into the horoball. Clearly, z_{2} ∈ P (Y). We will choose z to be of the form

z = (cos θ)u_{1} + (sin θ)z_{2}

for an appropriate choice of θ ∈ (0, \frac{π}{2} ]. Any such z will lie in 𝕊^{ℓ−1} ∗P (Y). Let ε = ⟨u_{2}, z_{2}⟩. Then ε > 0 and ⟨u, z⟩ = ⟨u_{1}, (cos θ)u_{1}⟩+⟨u_{2}, (sin θ)z_{2}⟩ = cos θ +ε sin θ. So, pick θ ∈ (0, \frac{π}{2} ] so that sin θ < ε. Then cos θ + ε sin θ > cos θ + sin^{2} θ > cos^{2} θ + sin^{2} θ = 1 and hence, ⟨u, z⟩ > 1. □

Lemma 3.2.8. Suppose v, w are points in P (X) with ⟨v, w⟩ > −1. Let u denote the orthogonal projection of w onto v^{⊥}, normalized to have unit length:

u = \frac{2 − ⟨v, w⟩ v}{(1 − ⟨v, w⟩^{2})^{\frac{1}{2}}} .

(So, u is the unit tangent vector to the circular arc in 𝕊_{1}^{n} from v to w.) Let Y be the face of X such that v belongs to the relative interior of σ_{Y}. Then there is a vector z in Lk(v, P (X)) (= P̃(Y, v^{⊥})) so that ⟨u, z⟩ ≥ 1. Moreover, if neither condition (i) nor (ii) of Theorem 3.2.2 holds, then z can be chosen so that ⟨u, z⟩ > 1.

Proof: Denote the orthogonal projections of v and w onto Span(Y) by v_{2} and w_{2}, respectively. Since v_{2} = 0, u_{2} = (1−⟨v, w⟩^{2})^{− \frac{1}{2}} w_{2}. For any x ∈ Y, 0 ≥ ⟨w, x⟩ = ⟨w_{1}, x⟩+⟨w_{2}, x⟩ = ⟨w_{2}, x⟩. Hence, w_{2} and its positive multiple u_{2} lie in C^{∗}(Y). Thus, we can apply Lemma 3.2.7 to find z ∈ P̃(Y) with ⟨u, z⟩ ≥ 1.

As for the question of strict inequality, it remains to show that cases (a), (b) and (c) of Lemma 3.2.7 correspond to situations (i) and (ii) of Theorem 3.2.2.

Case (a): u_{2} = 0. Then w_{2} = 0 and hence, w ∈ Y̌. Thus, both v and w belong to σ_{Y} and we are in situation (i).

Case (b): u_{2} is spacelike and d(G_{u_{2}}, Y) = 0. By Lemma 3.2.5 (ii), either G_{u_{2}} is a supporting hyperplane of Y, or G_{u_{2}} meets S_{∞}(Y) in a single point y. If G_{u_{2}} is a supporting hyperplane, then it intersects Y in a face Y′. For any x ∈ Y′, ⟨w, x⟩ = ⟨w_{2}, x⟩ = ⟨u_{2}, x⟩ = 0, so Y′ ⊂ H_{v} ∩ H_{w}, i.e., v, w both belong to σ_{Y′}. This is situation (i). Otherwise, H_{v} ∩ H_{w} ∩ X = ∅ and H_{v} ∩ H_{w} meets S_{∞}(X) at y. By Lemma 3.2.4, y is a cusp point of X. Hence, v, w ∈ P_{y} and we are in situation (ii).

Case (c): u_{2} is lightlike and represents a point y in S_{∞}(Y). We then argue exactly as in the end of the previous paragraph, to conclude that we are again in situation (ii). □

Proof of Theorem 3.2.2: Consider the function

f(v) = cos^{−1} ⟨v, w⟩

defined on the set

U = {v ∈ P (X) | ⟨v, w⟩ > −1}.

By Lemma 3.2.1, for any c > −1,

U_{c} = {v ∈ P (X) | ⟨v, w⟩ ≥ c}

is a compact subset of U.

Claim: For any v ∈ U there is a point r(v) ∈ U such that

d(v, r(v)) + f(r(v)) ≤ f(v)

and r(v) ≠ v if v ≠ w.

To prove the claim, suppose v ∈ U, v ≠ w. Let u and z be as in Lemma 3.2.8, i.e.,

u = \frac{w − ⟨v, w⟩ v}{(1 − ⟨v, w⟩^{2})^{\frac{1}{2}}} , and

⟨u, z⟩ ≥ 1, z ∈ Lk(v, P (X)).

For sufficiently small ε > 0, put

r(v) = (cos ε)v + (sin ε)z.

Let θ = cos^{−1} ⟨v, w⟩ so that w = (cos θ)v + (sin θ)u. Then

⟨r(v), w⟩ = cos ε cos θ + ⟨u, z⟩ sin ε sin θ

≥ cos ε cos θ + sin ε sin θ = cos(θ − ε).

Hence,

d(v, r(v)) + f(r(v)) ≤ ε + cos^{−1} ⟨r(v), w⟩
≤ ε + θ − ε
= f(v).

If neither condition (i) nor (ii) of the theorem holds, then we can choose z so that ⟨u, z⟩ > 1 (Lemma 3.2.8). Then ⟨r(v), w⟩ > cos(θ − ε), which gives a strict inequality, d(v, r(v)) + f(r(v)) < f(v).

Now fix v ∈ U and let V be the set

V = {x ∈ U | d(v, x) + f(x) ≤ d(v, r(v)) + f(r(v))}.

Note that for any x ∈ V, r(x) also lies in V since

d(v, r(x)) + f(r(x)) ≤ d(v, x) + d(x, r(x)) + f(r(x))
≤ d(v, x) + f(x).

Note also that V is contained in U_{⟨v,w⟩} since for x ∈ V

f(x) ≤ d(v, x) + f(x) ≤ d(v, r(v)) + f(r(v)) ≤ f(v).

Thus, V is nonempty (since r(v) ∈ V) and compact, so f attains its minimum on V, say at p ∈ V. We claim that p = w. For if not, then r(p) lies in V and since r(p) ≠ p,

0 < d(p, r(p)) ≤ f(p) − f(r(p)),

which contradicts the minimality of f at p. Hence, p = w. In particular, w ∈ V. Thus,

d(v, w) + f(w) ≤ d(v, r(v)) + f(r(v)) ≤ f(v)

with the last inequality strict except under conditions (i) or (ii). Since ⟨w, w⟩ = 1, f(w) = 0, so this proves the theorem.

3.3 Some consequences of Theorem 3.2.2. The goal of this subsection is to prove Proposition 3.3.3, below, which is the key technical lemma in the proof of the main result. We begin by recalling a general result concerning the orthogonal join of two piecewise spherical polyhedra.

Suppose that K_{1} and K_{2} are piecewise spherical polyhedra and that K_{1} ∗K_{2} denotes their orthogonal join (cf. 1.3). If x_{1} ∈ K_{1}, x_{2} ∈ K_{2}, then there is a geodesic segment (= arc of a great circle) of length \frac{π}{2} in K_{1} ∗ K_{2} from x_{1} to x_{2}. For t ∈ [0, \frac{π}{2} ], let [x_{1}, x_{2}, t] denote the point on this segment of distance t from x_{1}.

The following is proved in [CD1] as Lemma A2 and A6 of the Appendix.

Lemma 3.3.1. Suppose that K_{1}, K_{2} are nonempty piecewise spherical polyhedra and that d_{1}, d_{2} and d denote the intrinsic metrics on K_{1}, K_{2} and K_{1} ∗ K_{2}, respectively. Let x = [x_{1}, x_{2}, s] and y = [y_{1}, y_{2}, t] be points in K_{1} ∗ K_{2}. Then d(x, y) ≤ π. Moreover, d(x, y) = π if and only if s = t and one of the following conditions holds:

(a) s = 0 and d_{1}(x_{1}, y_{1}) ≥ π,

(b) s = \frac{π}{2} and d_{2}(x_{2}, y_{2}) ≥ π,

(c) s ≠ 0, s ≠ \frac{π}{2} , d_{1}(x_{1}, y_{1}) ≥ π and d_{2}(x_{2}, y_{2}) ≥ π.

In the next lemma and its proof we return to the notation of 3.2.6.

Lemma 3.3.2. Let Y be a convex polyhedral set in ℍ^{n−1} and let v, w ∈ P̃(Y) be such that d̃(v, w) ≥ π, where d̃ is the intrinsic metric on P̃(Y). Then

⟨v, w⟩ ≤ − 1.

In other words, the hyperbolic hyperplanes G_{v} = v^{⊥} ∩ ℍ^{n−1} and G_{w} = w^{⊥} ∩ ℍ^{n−1} are either parallel or ultraparallel.

Proof: If Y is of dimension n − 1 (the maximum possible), then P̃(Y) = P (Y) and the lemma follows immediately from Theorem 3.2.2. So, suppose dim Y < n − 1. For any vector v ∈ P̃(Y) let v_{1} and v_{2} denote its orthogonal projections onto Y^{⊥} (= ℝ^{ℓ}) and Span(Y), respectively. If q(v_{i}) ≠ 0, for i = 1, 2, let v̄_{i} = q(v_{i})^{− \frac{1}{2}} v_{i} be the corresponding unit vector. Thus, v̄_{1} ∈ 𝕊^{ℓ−1} and v̄_{2} ∈ P (Y). We want to apply the previous lemma to P̃(Y) = 𝕊^{ℓ−1} ∗ P (Y). In the notation of that lemma, v = [v̄_{1}, v̄_{2}, s] and w = [w̄_{1}, w̄_{2}, t], where s = q(v_{1})^{\frac{1}{2}} and t = q(w_{1})^{\frac{1}{2}} . Hence, Lemma 3.3.1 gives that d̃(v, w) = π, that s = t, and that one of the following conditions holds:

(a) v and w are antipodal points on the suspension sphere 𝕊^{ℓ−1},

(b) v and w lie in P (Y) and d(v, w) ≥ π, or

(c) q(v_{1}) ≠ 0, q(v_{2}) ≠ 0, v_{1} = −w_{1}, and d(v̄_{2}, w̄_{2}) ≥ π.

In case (a), ⟨v, w⟩ = −1. In case (b), ⟨v, w⟩ ≤ − 1, by Theorem 3.2.2. So, suppose (c) holds. Then ⟨v̄_{1}, w̄_{1}⟩ = −1 and (by Theorem 3.2.2) ⟨v̄_{2}, w̄_{2}⟩ ≤ − 1. Since s = t, q(v_{1}) = q(w_{1}) and q(v_{2}) = q(w_{2}). Hence, ⟨v_{1}, w_{1}⟩ = −q(v_{1}) and ⟨v_{2}, w_{2}⟩ ≤ − q(v_{2}). Thus, ⟨v, w⟩ = ⟨v_{1}, w_{1}⟩ + ⟨v_{2}, w_{2}⟩ ≤ − (q(v_{1}) + q(v_{2})) = −1, so the lemma also holds in this case. □

In the next proposition we return to the situation considered in 3.1.

Proposition 3.3.3. Let γ = (γ_{1}, …, γ_{k}) be a closed local geodesic in P (X) with break points v_{0}, …, v_{k−1}, v_{k} = v_{0}. (Interpret the indices as integers modulo k.) Let H_{i} = H_{v_{i}} and G_{i} = H_{i−1} ∩ H_{i} be the subspaces of ℍ^{n} considered in 3.1.

(i) The subspaces G_{i} and G_{i+1} are either parallel or ultraparallel hyperplanes in H_{i}.

(ii) For i = 0, …, k − 1, suppose that G_{i} and G_{i+1} are parallel. Let y_{i} ∈ S_{∞}(H_{i}) be the point at infinity where G_{i} and G_{i+1} meet. Further, suppose that y_{0} = y_{1} = ⋯ = y_{k−1}. Then ℓ(γ) = 2π and, if y denotes the common value of the y_{i}, then y is a cusp point of X.

Proof: (i) As in 1.5, from the broken geodesic γ we get unit tangent vectors, γ′_{i} and γ′_{i+1} in Lk(v_{i}, P (X)), where, in fact, γ′_{i} and γ′_{i+1} are the unit outward normals to G_{i} and G_{i+1} in H_{i}. Since γ is a local geodesic, condition (1.5.2) gives

d_{i}(γ′_{i}, γ′_{i+1}) ≥ π,

where d_{i} is the intrinsic metric on Lk(v_{i}, P (X)). Let Y_{i} be the face of X such that v_{i} belongs to the relative interior of σ_{Y_{i}}, i.e., Y_{i} = H_{i} ∩ X. By Lemma 2.6.2 (1), Lk(v_{i}, P (X)) = P̃(Y_{i}, (v_{i})^{⊥}). Applying the previous lemma with Y = Y_{i}, v = γ′_{i} and w = γ′_{i+1}, we conclude that (i) holds.

(ii) Suppose the hypotheses of (ii) hold. Let Z_{y} denote the intersection of the halfspaces bounded by the H_{i} with a horosphere E_{y} at y. Then Z_{y} is an (n − 1)-dimensional Euclidean prism (i.e., it is isometric to the Cartesian product of a polygon and an (n − 3)- dimensional Euclidean space), and ℓ(γ) is the sum of the exterior angles. Thus, ℓ(γ) = 2π. Moreover, if v, w are any two points on γ, then the length of the segment of γ between them is cos^{−1} ⟨v, w⟩. Choose such v, w which are arbitrarily close and which do not lie in the same cell of P (X). If y is not a cusp point, then by Theorem 3.2.2, d(v, w) < cos^{−1} ⟨v, w⟩, which contradicts the hypothesis that γ is a local geodesic. Hence, y is a cusp point and the lemma is proved.

4. The main result.

4.1 P (X) is large.

Theorem 4.1.1. Suppose X is a hyperbolic convex polyhedral set of dimension n. Then

1) Its polar dual P (X) is large,

2) If γ is any closed local geodesic of length 2π, then γ must lie in the subcomplex P_{y} for some cusp point y of X.

The proof uses Proposition 3.3.3, together with an argument of Rivin and Hodgson [RH].

Proof: According to Lemma 1.6.1, P (X) is large if and only if

(i) sys(P (X)) ≥ 2π,

(ii) sys(Lk(σ, P (X))) ≥ 2π for all cells σ in P (X).

Each cell σ of P (X) is of the form σ = σ_{Y} for some proper face Y of X. By Lemma 2.6.2 (2), Lk(σ_{Y} , P (X)) = P (Y). We can assume, by induction on the dimension, that the theorem holds for Y; in particular, sys(P (Y)) ≥ 2π, so condition (ii) holds. Thus, it suffices to check (i).

Let γ be a closed local geodesic in P (X). As in subsection 3.1, we construct an (n − 1)-dimensional hyperbolic cylinder Ŝ_{γ} and its image S_{γ} in ℍ^{n} so that ℓ(γ) is the sum of the exterior dihedral angles in S_{γ}. thus, Ŝ_{γ} is isometric to ℍ^{n}/ ⟨ρ_{γ}⟩ where the isometry ρ_{γ} is either hyperbolic or parabolic. So, there are two cases to consider.

Case 1: ρ_{γ} is hyperbolic. In this case the argument of [RH] shows that ℓ(γ) > 2π. (Actually, [RH] is concerned only with dimension 3, but this part of the argument works in any dimension.) The argument goes as follows. Since ρ_{γ} is hyperbolic, there is a unique closed geodesic α̂ on Ŝ_{γ} of shortest length (α̂ is the image of the axis of ρ_{γ}). Let α denote its image in S_{γ}. Then α is a piecewise geodesic in ℍ^{n}. The break points of α lie on the boundaries of the “strips” S_{i}, i.e., such a break point lies on the intersection of two hyperplanes H_{i} and H_{i+1}. By Lemma 3.2 in [RH], the “turning angle” of α at a break point is ≤ the corresponding exterior dihedral angle between H_{i} and H_{i+1}. Thus, the sum of the turning angles of α is ≤ ℓ(γ). By Theorem 3.1 in [RH], this sum of turning angles is > 2π. hence, ℓ(γ) > 2π.

Case 2: ρ_{γ} is parabolic. In this case we are in the situation of part (ii) of Proposition 3.3.3. The proposition implies that γ lies in some P_{y} for y a cusp point of X and its proof shows that ℓ(γ) = 2π.

Thus, ℓ(γ) ≥ 2π in both cases, which verifies condition (i) and hence, statement (1) of the theorem. The analysis in Case 2 also shows that if ℓ(γ) = 2π, then γ lies in some P_{y}, which is statement (2) of the theorem.

4.2 The induced metric at a cusp. We have the following converse to the last part of Theorem 3.2.2.

Proposition 4.2.1. Suppose X is a hyperbolic convex polyhedral set of dimension n and that d is the intrinsic metric on P (X). Suppose further that v, w are points in P (X) and that either (i) v and w belong to the same cell of P (X) or (ii) v and w belong to P_{y} for some cusp point y. Then

d(v, w) = cos^{−1} ⟨v, w⟩ .

Proof: We shall only verify this in case (ii), the proof in case (i) being similar and easier. Let d_{1} denote the intrinsic metric on P_{y}. We first note that, since P_{y} is the polar dual of a Euclidean convex set, if v, w ∈ P_{y}, then ⟨v, w⟩ ≥ − 1 and d_{1}(v, w) = cos^{−1} ⟨v, w⟩ (by Lemma 2.2.1 (2)). The proof is by induction on n. So, suppose the proposition holds in dimensions < n. We first claim that P_{y} is locally convex in P (X). To see this, we must check condition (∗) of Lemma 1.6.5. Let u ∈ P_{y} and Y the face of X such that u belongs to the relative interior of σ_{Y}. Then

Lk(u, P (X)) = P̃(Y, u^{⊥}) = 𝕊^{ℓ−1} ∗ P (Y)
Lk(u, P_{y}) = 𝕊^{ℓ−1} ∗ P_{y}(Y)

where P_{y}(Y) = P_{y} ∩ Y and 𝕊^{ℓ−1} is the unit sphere in Y^{⊥} ∩ u^{⊥}. Let d′ and d′_{1} denote the intrinsic metrics on P (Y) and P_{y}(Y), respectively. Suppose we are given x, x′ ∈ P_{y}(Y) such that d′_{1}(x, x′) ≥ π. Then d′_{1}(x, x′) = π and ⟨x, x′⟩ = −1. By the inductive hypothesis, d′(x, x′) = π. By Lemma 3.3.1, the same result holds after taking ℓ-fold suspensions. This verifies condition (∗) of Lemma 1.6.5 and hence, shows that P_{y} is locally convex in P (X). By Lemma 1.6.4, any geodesic in P_{y} is actually geodesic in P (X). Hence,

cos^{−1} ⟨v, w⟩ = d_{1}(v, w) = d(v, w).

Definition 4.2.2. Let Cone(P_{y}) denote the orthogonal join of P_{y} with a point. The completed polar dual of X, denoted P̂(X), is the piecewise spherical complex formed by gluing Cone(P_{y}) to P (X) along P_{y} for each cusp point y.

For example, if S is n-dimensional and of finite volume, then each P_{y} is isometric to the round sphere 𝕊^{n−2} (by Lemma 2.2.1 (3)), so Cone(P_{y}) is a hemisphere. Thus, P̂(X) is obtained from P (X) by “capping off” the P_{y} with hemispheres. It is then clear that P̂(X) is homeomorphic to the (n − 1)-sphere.

In general, if K_{1}, K_{2} are large piecewise spherical subcomplexes, and K_{0} is a common subcomplex which is locally convex in both, then the result of gluing K_{1} to K_{2} along K_{0} is large. (After taking Euclidean cones on K_{1}, K_{2} and K_{0}, this is a special case of Gromov’s Gluing Lemma; see [Pa, Lemma 4.3].) As a corollary of Proposition 4.2.1 and Theorem 4.1.1, we therefore have the following.

Corollary 4.2.3. P̂(X) is large.

4.3 Convex polytopes. In this subsection we state two special cases of the above results.

Corollary 4.3.1. Let X be an n-dimensional hyperbolic convex polytope. Then P (X) is a large piecewise spherical structure on 𝕊^{n−1}. Moreover, the systole of P (X) is greater than 2π, as is the systole of the link of any cell in P (X).

Corollary 4.3.2. Let X be an n-dimensional convex polyhedral set of finite volume in ℍ^{n}. Then P̂(X) is a large, piecewise spherical structure on 𝕊^{n−1}.

5. The Gauss-Bonnet Formula for hyperbolic polytopes.

5.1 The quantity κ(K). For σ a p-dimensional spherical cell, let a(σ) denote its p-dimensional volume, normalized so that the volume of 𝕊^{p} is 1, i.e.,

a(σ) = \frac{vol(σ)}{vol(𝕊^{p})}.

Also, let a^{∗}(σ) = a(σ^{∗}), where σ^{∗} = C^{∗}(σ) ∩ S^{p} is the dual cell to σ (cf. 2.3.1). Given a finite, piecewise spherical cell complex K, consider the following quantity:

(5.1.1)  κ(K) = 1 + ∑_{σ}(−1)^{dim σ+1}a^{∗}(σ),

where the summation is over all cells σ of K. In [CMS] it is shown that the value of κ on the link of a vertex in a piecewise Euclidean complex plays the role of the Gauss-Bonnet integrand. For this reason, in analogy with a well-known conjecture of Hopf, we made the following conjecture in [CD2].

Conjecture 5.1.2. Suppose that K is a large, piecewise spherical structure on the (2m − 1)-sphere. Then (−1)^{m}κ(K) ≥ 0.

5.2 Hopf’s formula. Suppose X^{n} is an n-dimensional hyperbolic convex polytope. Its normalized volume v(X^{n}) is defined by

v(X^{n}) = \frac{vol(X^{n})}{vol(𝕊^{n})}.

Thus, v(X^{n}) is a positive constant times the usual volume of X^{n}.

The Gauss-Bonnet Formula for hyperbolic convex polytopes is due to H. Hopf [H]. (Of course, Hopf’s result predates the general Gauss-Bonnet Formula due to Allendoerfer and Weil [AW].) Hopf’s formula is the following.

Theorem 5.2.1 (Hopf). Suppose X^{2m} is a hyperbolic convex polytope of dimension 2m. Then

(−1)^{m}2v(X) = κ(P (X)).

Remarks 5.2.3. (1) The analogous formula for X^{2m} spherical or Euclidean was proved by Poincaré in 1905, [P]. It states that if X is Euclidean then κ(P (X)) = 0 (this also holds in odd dimensions), while if X^{2m} is a spherical cell, then 2a(X) = κ(P (X)).

(2) In writing the quantity κ(P (X)), we have dualized twice: once in forming P (X) and a second time in taking the normalized “exterior angles” a^{∗}(σ). Thus, κ(P (X)) could have been written as an alternating sum of interior angles of X (and this is the actual form of the formulas of Poincaré and Hopf).

(3) If we subdivide X^{2m} into 2m-dimensional cells, X_{1}, …, X_{k}, then a formal calculation shows that κ(P (X)) = ∑ κ(P (X_{i})). (Compare Theorem III in [AW].) The left-hand side of Hopf’s formula is also clearly additive with respect to subdivisions. Thus, it suffices to prove the formula in the case of a simplex.

(4) The formula also holds when X is a convex polyhedral set of finite volume (use a limit argument). In this case it does not matter whether we write the right-hand side as κ(P (X)) or as κ(P̂(X)). The reason is that for any cusp point y, κ(P_{y}) = 0 (by Poincaré’s result) and hence, κ(Cone(P_{y})) = 0 (since κ is multiplicative for orthogonal joins, by [CMS], formula (3.2.9). Thus, κ(P (X)) = κ(P̂(X)).

Since v(X^{2m}) > 0, we have the following corollary.

Corollary 5.2.3. Let X^{2m} be a convex polyhedral set of finite volume in ℍ^{2m} (e.g., a convex polytope). Then Conjecture 5.1.2 holds for P̂(X). In fact, we have a strict inequality (−1)^{m}κ(P̂(X)) > 0.

6. Spaces of geometric structures.

6.1 The space of piecewise spherical structures. Let K be a finite simplicial complex. Denote by K^{(i)} the set of i-simplices in K. A piecewise spherical structure on K is determined by a function ℓ : K^{(1)} → (0, π), written as e → ℓ_{e}, giving the edge lengths. Conversely, such a function comes from a piecewise spherical structure on K if for each simplex σ in K of dimension > 1 there is a spherical simplex with edge lengths as prescribed by ℓ.

This last condition is equivalent to a simple condition using linear algebra, which we shall now describe. Given ℓ : K^{(1)} → (0, π) define a new function x : K^{(i)} → (−1, 1) by x_{e} = cos(ℓ_{e}). Clearly, ℓ and x determine one another. Suppose that σ ∈ K^{(m)}, m > 1, and that {v_{1}, ⋯, v_{m+1}} is the vertex set of σ. Given an arbitrary function x : K^{(1)} → (−1, 1), define a (m + 1) × (m + 1) symmetric matrix c_{σ}(x) by

c_{σ}(x)_{ij} = \{ \begin{matrix}1 & , i = j \\ x\overline{v_{i}v_{j}} & , i ≠ j \end{matrix}

where \overline{v_{i}v_{j}} denotes the edge from v_{i} to v_{j}. Then there exists a spherical m-simplex with edge lengths as prescribed by x if and only if c_{σ}(x) is positive definite. (If σ can actually be realized as a m-simplex in 𝕊^{m}, then v_{1}, …, v_{m+1} are unit vectors in ℝ^{m+1} and c_{σ}(x) is the matrix of inner products (v_{i}, v_{j}).)

The space PS_{K} of piecewise spherical structures on K is defined to be the subset of the Euclidean space ℝ^{K^{(1)}} consisting of all functions x : K^{(1)} → (−1, 1) such that c_{σ}(x) is positive definite for all simplices σ in K (of dimension > 1).

Lemma 6.1.1 PS_{K} is a convex open subset of ℝ^{K^{(1)}}.

Proof: That the c_{σ}(x) be positive definite is clearly an open condition. A convex linear combination of positive definite matrices is positive definite; hence, PS_{K} is a convex subset of ℝ^{K^{(1)}}. □

Given x ∈ PS_{K}, let K_{x} denote the space K with the piecewise spherical structure determined by x. Similarly, if σ is a simplex in K, then σ_{x} denotes the corresponding spherical simplex determined by x.

A point x ∈ PS_{K} is extra large if the following two conditions hold:

a) sys (K_{x}) > 2π

b) sys (Lk(σ_{x}, K_{x})) > 2π for all simplices σ in K. (See §1.6 for the definition of “sys”.) Let EL_{K} denote the set of those x in PS_{K} which are extra large.

Moussong showed in [M, Lemma 5.11] that the function sys : PS_{K} → ℝ defined by x ↦ sys (K_{x}) is lower semicontinuous. This gives the following result.

Lemma 6.1.2 (Moussong) EL_{K} is an open subset of PS_{K}.

If X is a convex polyhedral set in hyperbolic space, without cusp points, then our main result (Theorem 4.1.1) asserts that P (X) is extra large. Hence, whenever K is combinatorially equivalent to the polar dual of such an X, the space EL_{K} is nonempty.

6.2 Spaces of polytopes. Suppose that Q is some n-dimensional convex simplicial polytope. (Simplicial means that each proper face of Q is a simplex). Since Q is simplicial, its dual polytope Q^{∗} is called simple. We are only interested in Q and Q^{∗} up to combinatorial equivalence. The boundary complex of Q is a simplicial complex which we shall denote by K. (Thus, K is P L-homeomorphic to 𝕊^{n−1}.)

Let H_{K} (respectively, S_{K} or E_{K}) denote the space of isometry classes of hyperbolic (respectively, spherical or Euclidean) polytopes which are combinatorially equivalent to Q^{∗}. The topology on these spaces will be described in the course of the proof of the next lemma.

Lemma 6.2.1 The spaces H_{K}, S_{K}, and E_{K} are naturally smooth manifolds of dimension nf_{0} − ( \begin{matrix}n+1 \\ 2 \end{matrix} ), where f_{0} denotes the number of vertices in K.

Proof: Suppose that X is a convex n-cell in ℍ^{n} combinatorially isomorphic to Q^{∗}. For each vertex v_{i} of K we have a codimension one face of X and a outward pointing unit normal u_{i} in the unit pseudo-sphere 𝕊_{1}(ℝ^{n,1}). Thus, X determines an f_{0}-tuple u : K^{(0)} → 𝕊_{1}(ℝ^{n,1}) of vectors in 𝕊_{1}(ℝ^{n,1}).

A small neighborhood of u in [S_{1}(ℝ^{n,1})]^{f_{0}} will determine a polytope of the same combinatorial type (since X is simple). Since dim 𝕊_{1}(ℝ^{n,1}) = n, this shows that the space of such X is a manifold of dimension nf_{0}. The isometry group O(n, 1) of ℍ^{n} acts properly and freely on this manifold and H_{K} is the quotient manifold. Since dim O(n, 1) = ( \begin{matrix}n+1 \\ 2 \end{matrix} ),

we conclude that dim H_{K} = nf_{0}−( \begin{matrix}n+1 \\ 2 \end{matrix} ). The analysis for S_{K} and E_{K} is entirely similar. □

Remark Actually H_{K} should be called the space of “marked” hyperbolic polytopes of the same combinatorial type as Q^{∗} (since we have prescribed an identification of the codimension one faces of X with the vertices of K). The finite group G of combinatorial symmetries of Q^{∗} acts on H_{K} and H_{K}/G is the “unmarked” space. Similar remarks apply to S_{K} and E_{K}.

If X represents an element of H_{K} (respectively, S_{K} or E_{K}), then P (X) represents an element of PS_{K}. This defines a map P : H_{K} → PS_{K} (respectively, P : S_{K} → PS_{K} or P : E_{K} → PS_{K}). The following result for n = 3 is proved as Corollary 4.6 of [RH]. Another 3-dimensional result, Theorem 4.1 of [RH], immediately implies that both results hold in all dimensions ≥ 3.

Proposition 6.2.2 ([RH]) For n ≥ 3, the maps P : H_{K} → PS_{K} and P : S_{K} → PS_{K} are embeddings.

The map P : E_{K} → PS_{K} is not an embedding. To see this, first note that the elements of P (E_{K}) are all isometric to the round sphere 𝕊^{n−1}. Thus, the points in P (E_{K}) may be regarded as geodesic triangulations of 𝕊^{n−1} of the combinatorial type of K. Conversely, any such geodesic triangulation is clearly in P (E_{K}). (If {v_{1}, …, v_{k}} ⊂ 𝕊^{n−1} is the vertex set of such a triangulation, then the Euclidean n-cell X defined as {z ∈ ℝ^{n} | ⟨z, v_{i}⟩ ≤ 1, 1 ≤ i ≤ k} gives back the geodesic triangulation as its polar dual.) Arguing as in Lemma 6.2.1, we see that P (E_{K}) is a submanifold of dimension (n − 1)f_{0} − \binom{n}{2}. Hence, the fiber of P : E_{K} → PS_{K} has dimension f_{0} − n.

Lemma 6.2.3 P (E_{K}) ⊂ \overline{P (H_{K})} ∩ \overline{P (S_{K})}.

Proof: Realize a point in P (E_{K}) as a geodesic triangulation of 𝕊^{n−1}. Identify 𝕊^{n−1} with a codimension one sphere in the unit pseudo-sphere 𝕊_{1}^{n}(ℝ^{n,1}), e.g., let z ∈ ℍ^{n} and identify 𝕊^{n−1} with the intersection of z^{⊥} and 𝕊_{1}^{n}(ℝ^{n,1}). Push the vertices of the triangulation slightly into the “half-pseudo-sphere” defined by v·z < 0. If u_{1}, …, u_{k} are the resulting unit vectors in 𝕊_{1}^{n}(ℝ^{n,1}), then the hyperbolic polytope X defined by X = {w ∈ ℍ^{n} | u_{i} · w ≤ 0} has P (X) combinatorially equivalent to K and close to the original triangulation of 𝕊^{n−1}. Thus, P (E_{K}) ⊂ \overline{P (H_{K})}. The proof that P (E_{K}) ⊂ \overline{P (S_{K})} is similar, with the pseudosphere 𝕊_{1}^{n}(ℝ^{n,1}) replaced by the unit sphere 𝕊^{n} in ℝ^{n+1}. □

Thus, when K is the boundary complex of a simplicial polytope, the following picture has emerged. The space PS_{K} of piecewise spherical structures on K is diffeomorphic to Euclidean space of dimension f_{1}, where f_{1} denotes the number of edges in K. PS_{K} contains submanifolds P (H_{K}) and P (S_{K}) each of dimension nf_{0} − ( \begin{matrix}n+1 \\ 2 \end{matrix} ). The fact that P (H_{K}) is nonempty means that the open subset EL_{K} of extra large structures is nonempty. The fact that P (E_{K}) ⊂ \overline{P (H_{K})} means that any geodesic triangulation of the round sphere can be deformed to an extra large piecewise spherical structure.

An interesting problem is to understand the topology of EL_{K}. For example, when is EL_{K} connected?

6.3 The Lower Bound Theorem. Since P (H_{K}) is a submanifold of PS_{K}, we have

dim P (H_{K}) ≤ dim PS_{K}.

In the combinatorial theory of convex polytopes, this fact is known as “the Lower Bound Theorem” which we state below.

Theorem 6.3.1 (Barnette [Ba]) Let Q be an n-dimensional simplicial polytope, n ≥ 3, with f_{0} vertices and f_{1} edges. Then

f_{1} ≥ nf_{0} − ( \begin{matrix}n+1 \\ 2 \end{matrix} ) .

For n > 3, the above inequality is strict unless Q is a so called “stacked polytope”. This means that K (= ∂Q) is obtained from the boundary of an n-simplex via subdivisions which involve adjoining barycenters of (n-1)-simplicies. Thus, if n > 3 and Q is not stacked, then P (H_{K}) is a proper subset of EL_{K}.

Some explicit examples of elements in EL_{K} − P (H_{K}) are given in [M]. One such class of examples occurs when Q^{∗} is the product of two 2-simplicies (so that K is the join of two triangles). Realize Q^{∗} as the Cartesian product of two 2-simplicies in ℍ^{2} with acute angles. Its polar dual is a piecewise spherical structure x on K which is the orthogonal join of two large triangles. It follows that x is large. Consider the 6 × 6 matrix c(x) obtained by taking inner products of the 6 unit vectors in ℝ^{2,1} × ℝ^{2,1} which are normal to the codimension-one faces of Q^{∗}. Thus, c(x) has signature (4,2). (Alternately described, c(x) is the matrix of cosines of edge lengths in the piecewise spherical structure x.) There are 9 edges of K of length π/2. If we deform x to x′ by slightly increasing these edge lengths, then it follows from Moussongs Theorem that x′ is extra large. On the other hand, for small deformations, the matrix c(x′) will still have signature (4,2). Hence, x′ cannot arise as a polar dual of a hyperbolic polyhedra (since this would require c(x′) to be of signature (5,1)).

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