Isometry-Invariant Valuations on Hyperbolic Space

Discrete & Computational Geometry, Aug 2006

Daniel A. Klain

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Isometry-Invariant Valuations on Hyperbolic Space

Discrete Comput Geom Geometry Discrete & Computational Daniel A. Klain 0 1 0 Department of Mathematical Sciences, University of Massachusetts Lowell , Lowell, MA 01854 , USA 1 Isometry-Invariant Valuations on Hyperbolic Space Hyperbolic area is characterized as the unique continuous isometry-invariant simple valuation on convex polygons in H2. We then show that continuous isometryinvariant simple valuations on polytopes in H2n+1 for n ≥ 1 are determined uniquely by their values at ideal simplices. The proofs exploit a connection between valuation theory in hyperbolic space and an analogous theory on the Euclidean sphere. These results lead to characterizations of continuous isometry-invariant valuations on convex polytopes and convex bodies in the hyperbolic plane H2, a partial characterization in H3, and a mechanism for deriving many fundamental theorems of hyperbolic integral geometry, including kinematic formulas, containment theorems, and isoperimetric and Bonnesen-type inequalities. - 0. Introduction A valuation on polytopes, convex bodies, or more general class of sets, is a finitely additive signed measure; that is, a signed measure that may not behave well (or even be defined) when evaluated on infinite unions, intersections, or differences. A more precise definition is given in the next section. Examples of isometry-invariant valuations on Euclidean space include the Euler characteristic, mean width, surface area, and volume (Lebesgue measure) [KR], [McM3]. Other important valuations on convex bodies and polytopes include projection functions and cross-section measures [Ga], [KR], [Sc1], affine surface area [Lut2], [Lut1], and Dehn invariants [Sah]. Unlike the countably additive measures of classical analysis, which are easily characterized using well-established tools such as the total variation norm, Jordan decomposition, and the Riesz representation theorem [Ru], valuations form a more general class of set functionals that has so far resisted such sweeping classifications [KR], [McM3]. The study of valuations on hyperbolic polytopes is motivated in part by the characterization of many classes of valuations on polytopes and compact convex sets in ∗ This research was supported in part by NSF Grant #DMS-9803571. Euclidean space. Such characterizations have had fundamental impact in convex, integral, and combinatorial geometry [Al1]–[Al3], [Ha], [KR], [Kl2], [Kl3], [Lud], [LR], [Sc2], [McM1]–[McM3] as well as to the theory of dissection of polytopes [Bo], [Ha], [KR], [McM3], [Sah]. The fundamental theorem of invariant valuation theory, Hadwiger’s characterization theorem, classifies all continuous isometry-invariant valuations on convex bodies in Rn as consisting of the linear span of the quermassintegrals (or, equivalently, of McMullen’s intrinsic volumes [McM3]): Theorem 0.1 (Hadwiger). Suppose that ϕ is a continuous valuation on compact convex sets in Rn, and that ϕ is invariant under Euclidean isometries. Then there exist c0, c1, . . . , cn ∈ R such that ϕ(K ) = n i=0 ci Vi (K ), for all compact convex sets K ⊆ Rn. In particular, if ϕ vanishes on sets K having dimension less than n, then ϕ is proportional to n-dimensional Euclidean volume Vn. Here the functionals Vi denote extensions of i -dimensional volume to continuous valuations on bodies in Rn, with suitable normalizing constants, so that each Vi is equal to i -volume when restricted to i -dimensional flats in Rn. In particular, Vn denotes volume in Rn, Vn−1 is one-half of the surface area, and so on down to the (renormalized) mean width V1 and the Euler characteristic V0. Hadwiger presented this theorem in [Ha]; alternative shorter proofs can be found in [Kl1] and [KR]. There remain a great many questions regarding how aspects of the Brunn–Minkowski theory of convex bodies (and polytopes) in Euclidean space can be extended to spaces having curvature. Even for spaces of constant curvature, such as the sphere and hyperbolic space, there are many unanswered questions, although work is being done to fill the gap (see, for example, [Fu], [GHS1], [GHS2], and [Ho]). In particular, little is yet known about invariant valuations on polytopes and convex sets in non-Euclidean spaces. A spherical analogue of Theorem 0.1, while plausible for the n-sphere Sn, remains an open question for n ≥ 3 (see, for example, [KR]). It also remains an open question whether a version of Theorem 0.1 holds for valuations defined only on polytopes in Euclidean space (as opposed to the larger class of compact convex sets) [McM3], [MS]. The present article characterizes hyperbolic area as the unique continuous isometryinvariant valuation on hyperbolic polygons that vanishes when restricted to points and lines. While in the hyperbolic plane H2 all ideal triangles (triangles with all three vertices at infinity) are isometrically congruent, this is not true for ideal simplices in higherdimensional hyperbolic spaces. In odd-dimensional spaces H2n+1 we show that hyperbolic volume is the unique continuous isometry-invariant valuation on hyperbolic polygons that vanishes on lower-dimensional polytopes and agrees with volume on all ideal simplices (i.e., simplices having all vertices at infinity). More precisely, we show that any continuous isometry-invariant simple valuation (i.e., vanishing on lower-dimensional sets) is determined uniquely by its values on ideal simplices. It is assumed throughout that the valuations in question are defined on all compact polyhedra in Hn as well as those having a finite set of vertices at infinity, although valuations are permitted to take infinite values on non-compact sets. These theorems then provide partial analogues to Hadwiger’s Theorem 0.1 for the hyperbolic plane H2. In analogy to the Euclidean case [Ha], [KR], [San], we also indicate briefly some consequences of valuation characterizations to integral geometry in hyperbolic space. In particular, the principal kinematic formula [KR], [San] for the Euler characteristic of a random intersection is generalized to a kinematic formula for arbitrary isometry-invariant valuations on H2. The main theorems of this article are indexed as follows: Valuation characterization theorems: Theorem 2.1. Area in H2. Theorem 3.2. Continuous invariant valuations on H2. Theorem 5.3. Finite invariant simple valuations on H2n+1. Theorem 5.5. Continuous invariant simple valuations on H2n+1. Corollary 5.6. Continuous invariant valuations on H3. Kinematic formulas and other consequences: Corollary 4.1. Valuation proof of the Gauss–Bonnet formula in H2. Theorem 4.3. Kinematic formula for continuous invariant valuations on H2. Corollary 4.4. Principal kinematic formula for H2. Corollary 4.5. Area formula for parallel bodies in H2. 1. Convexity in Hyperbolic Space Let Hn denote n-dimensional hyperbolic space; that is, the open upper half-space of Rn, Hn = {(x1, . . . , xn) | xn > 0} endowed with the hyperbolic distance metric. Recall that hyperplanes (flats) in Hn correspond to Euclidean hemispheres and half-hyperplanes that are orthogonal, in the Euclidean sense, to Rn−1. See, for example, either of [An] or [St]. We denote by Dn the disk (n-ball) model of hyperbolic space. (See again [An] or [St].) A set P ⊆ Hn is a convex polytope if P can be expressed as a finite intersection of half-spaces so that P is either compact or has at most a finite number of points (vertices) that lie at infinity (on the boundary of Hn or Dn). Denote by P(Hn) the set of all convex polytopes in Hn. A polytope is a finite union of convex polytopes. More generally, a set K ⊆ Hn is called convex if any two points of K can be connected by a hyperbolic line segment inside K . Let K(Hn) denote the union of P(Hn) with the set of all compact convex sets in Hn. Note that the only non-compact sets in K(Hn) are the convex polytopes having a finite collection of vertices at infinity. If K ∈ K(Hn) is non-compact, then K can be expressed in the form K = C ∗ L = convex hull of C ∪ L , where C ⊆ Hn is a compact convex set and L ⊆ ∂Hn is a finite set of points at infinity. Let S be a family of subsets of Hn closed under intersection and containing the empty set ∅. A function ϕ : S −→ R ∪ {±∞} is called a valuation on S if, for K , L ∈ S: ( 1 ) ϕ(∅) = 0. ( 2 ) ϕ(K ) ∈ R (that is, ϕ(K ) = ±∞) whenever K is compact. ( 3 ) If K ∪ L ∈ S as well, and if at least three of the values ϕ(K ), ϕ(L), ϕ(K ∩ L), and ϕ(K ∪ L) are finite, then ϕ(K ∪ L) + ϕ(K ∩ L) = ϕ(K ) + ϕ(L). This third condition is known as the inclusion–exclusion identity for valuations. A valuation ϕ on S is called invariant if ϕ(g K ) = ϕ(K ) for all isometries g of Hn such that K , g K ∈ S. A valuation ϕ on S is called finite if ϕ(K ) ∈ R (that is, ϕ(K ) = ±∞) for all K ∈ S. A valuation ϕ on a family of sets in Hn is simple if ϕ vanishes on sets of dimension strictly less than n. We apply the notion of Hausdorff topology on compact subsets of Hn with respect to hyperbolic distance. Definition 1.1. If Cm ∈ K(Hn) is a sequence of compact sets, and if limm→∞ Cm = C in the Hausdorff topology, then we say Cm → C . More generally, suppose that Cm ∈ K(Hn) is a sequence of compact sets and that L is finite subset of ∂Hn. Let Km = Cm ∗ L, the convex hull. Then we say that Km → K iff Cm → C . Using this definition we can now define continuous valuations on P(Hn) and K(Hn). Definition 1.2. A real-valued function ϕ on P(Hn) (resp. K(Hn)) is called continuous iff ϕ(K ) = ml→im∞ ϕ(Km ), whenever Ki → K in P(Hn) (resp. K(Hn)). The following is an adaptation of a theorem of Groemer [Gr] for Hn and for the Euclidean sphere Sn. Theorem 1.3 (Groemer’s Extension Theorem). (i) A valuation ϕ defined on convex polytopes in Hn (resp. Sn) admits a unique extension to a valuation on the lattice of all polytopes in Hn (resp. Sn). (ii) A continuous valuation ϕ on compact convex sets in Hn (resp. Sn) admits a unique extension to a valuation on the lattice of finite unions of compact convex sets in Hn (resp. Sn). In each case the extension of ϕ to finite unions is defined by suitable iteration of the inclusion–exclusion identity. The proof of Theorem 1.3 for the Euclidean case (Groemer’s original result) goes through for Hn and Sn without essential change, since the original proof is based not on the geometry of Rn, but rather on the algebra of indicator functions and the fact that a polytope is the intersection of half-spaces in Rn, a property that carries over analogously to convex polytopes in Hn and Sn as well. For the details of Groemer’s original proof, see [Gr] or [KR, p. 44]. In the arguments that follow, the unique extension of valuation ϕ given by Theorem 1.3 allows us to consider the value of ϕ on all finite unions of convex polytopes (or compact convex sets in Hn or Sn), whether or not such unions are actually convex. Moreover, Definition 1.2 allows us to consider the continuity of valuations at noncompact polytopes having vertices at infinity. Note, however, that only converging sequences of convex polytopes (or bodies) are considered. A continuous valuation on P(Hn) (resp. K(Hn)) may not necessarily respect convergent sequences of non-convex sets. (Surface area and perimeter generally do not, for example, even in the Euclidean context.) 2. A Characterization for Hyperbolic Area We now turn our attention to the two-dimensional space H2. Important examples of continuous invariant valuations on P(H2) and K(H2) include hyperbolic area, denoted A, hyperbolic perimeter P, the Euler characteristic χ , and a related functional χ∞, to be defined later. The perimeter P(K ) of a convex region K with non-empty interior in H2 is given by the length of the boundary ∂ K in the hyperbolic metric. If K is a one-dimensional convex set (i.e., a line segment) then P(K ) is equal to twice the length of K . This normalization guarantees that the perimeter functional is a continuous valuation on K(H2). The Euler characteristic χ (K ) of a compact set K ∈ K(Hn) is defined by χ (K ) = 1 if K = ∅, while χ (∅) = 0. Theorem 1.3 ensures that χ has a unique continuous and invariant extension to all finite unions of compact convex sets in K(Hn). This extension agrees with the usual definition of Euler characteristic on cell complexes [Mu, p. 124]. In particular, it follows that a compact polygon P decomposed into f0 vertices, f1 edges, and f2 triangles has Euler characteristic χ = f0 − f1 + f2. We can extend χ to non-compact sets K ∈ K(Hn) as follows. If a convex set K ∈ K(Hn) has exactly m points at infinity, that is, m limit points on ∂Hn, then define χ (K ) = 1 − m. Once again χ extends to a valuation on finite unions of (possibly non-compact) sets in K ∈ K(Hn) This extension of χ to polytopes having vertices at infinity is not unique. We will always denote by χ the extension just given. All other possibilities are accounted for by introducing a related (but distinct) invariant valuation χ∞ that is special to the hyperbolic case (and does not appear in the Euclidean or spherical contexts). This functional is defined and used in Section 3. It has been shown that every continuous invariant valuation on polygons in the Euclidean plane (or 2-sphere) is a linear combination of the valuations χ , P, and A (see [Ha] for the Euclidean case or [KR, p. 156] for both the Euclidean and spherical cases). In Section 3 we prove an analogous theorem for the hyperbolic plane: that every continuous invariant valuation on P (H2) (or K(H2)) is a linear combination of the valuations χ , χ∞, P, and A. As a fundamental step towards the results of Section 3, we first prove a characterization theorem for hyperbolic area. Recall that a valuation ϕ on P (H2) is simple if ϕ vanishes on points and on all one-dimensional sets. Theorem 2.1 (Area Characterization Theorem). Suppose that ϕ is a continuous invariant simple valuation on P (H2). Then there exists c ∈ R such that ϕ(K ) = c A(K ), for all K ∈ P (H2). It will be seen that continuity is a necessary condition for Theorem 2.1 to hold (see note after Proposition 2.3). Before the proof of Theorem 2.1, we consider some preliminary cases. Proposition 2.2. Suppose that ϕ is a continuous invariant simple valuation on P (H2). Then ϕ is finite on P (H2). Proof. From the definition of valuations we know that ϕ(K ) is finite whenever K is compact. Suppose that T is a triangle in H2 having exactly one vertex at infinity, such as AC X in Fig. 1, and suppose that ϕ(T ) = ±∞. Let M denote the midpoint of A and C . Note that AC X = A M X ∪ M C X , while A M X ∩ M C X = M X , a hyperbolic ray. Suppose that ϕ( A M X ) and ϕ( M C X ) are both finite. Because ϕ is simple, we have ϕ(M X ) = 0. The inclusion–exclusion identity would then yield ϕ(T ) = ϕ( AC X ) = ϕ( A M X ) + ϕ( M C X ) − ϕ(M X ), a finite value, contradicting our assumption that ϕ(T ) = ±∞. In other words, if ϕ(T ) = ±∞ then either ϕ( A M X ) = ±∞ or ϕ( M C X ) = ±∞ as well. Without loss of generality suppose that ϕ( A M X ) = ±∞. Iterating this procedure yields a sequence of points M1, M2, . . . → A such that ϕ( A Mi X ) = ±∞. Since these triangles converge to the hyperbolic ray A X = A A X , the continuity of ϕ implies that ϕ( A X ) = ±∞. However, this contradicts the fact that ϕ is a simple valuation, which requires that ϕ( A X ) = 0. It follows that ϕ(T ) = ±∞. We have shown that if T is a triangle in H2 having exactly one vertex at infinity, then ϕ(T ) is finite. More generally, if K is a polygon with (some or all) vertices at infinity, then K can be expressed as a finite union of triangles each having at most one vertex at infinity (using barycentric subdivision, for example). Since ϕ is simple, ϕ(K ) is equal to the sum of ϕ over these triangular pieces, so that ϕ(K ) is also finite. The next proposition is elementary (although useful). Proposition 2.3. Suppose that ϕ is an invariant simple valuation on closed line segments in R, and suppose that ϕ([α, β]) = 0 for some β > α ∈ R. Then ϕ(L) = 0 for all closed line segments L ⊆ R having length q(β − α), where q is any non-negative rational number. If ϕ is also continuous, then ϕ(L) = 0 for all closed line segments L ⊆ R. A hyperbolic triangle I is called ideal if its three distinct vertices all lie on the line at infinity, or, equivalently, if I has non-empty interior and its three angles each measure zero. Ideal polygons in H2 are defined similarly. Recall that all ideal hyperbolic triangles in H2 are isometrically congruent [St, p. 100]. A hyperbolic triangle S is called semi-ideal if two of its three vertices lie on the line at infinity. If a semi-ideal triangle S has single non-zero angle θ , then we typically denote it by Sθ . Recall that two semi-ideal hyperbolic triangles in H2 having the same non-zero angle θ are isometrically congruent. Proposition 2.4. Suppose that ϕ is an invariant simple valuation on P(H2), and suppose that ϕ(Sθ ) = 0 for all semi-ideal triangles Sθ ⊆ H2. Then ϕ(T ) = 0 for all triangles T ∈ P(H2). Proof. To begin, suppose that a hyperbolic triangle T has one vertex at infinity. (See, for example, AC X in Fig. 1.) In this case T can be expressed in terms of two semi-ideal triangles S and S , namely T ∪ S = S . For example, AC X ∪ C Y X = AY X in Fig. 1. Since ϕ is a simple valuation (so that ϕ vanishes on edges and vertices), we have ϕ(T ∩ S) = 0, so that ϕ(T ) = ϕ(T ) + 0 = ϕ(T ) + ϕ(S) = ϕ(T ∪ S) + ϕ(T ∩ S) = ϕ(S ) + 0 = 0. More generally, consider a typical hyperbolic triangle T . In this case T can be expressed in terms of triangles T1, T2, each having at least one vertex at infinity; that is, T ∪ T1 = T2. For example, A BC ∪ AC X = A B X in Fig. 1. Since ϕ(T1) = ϕ(T2) = 0, it follows that ϕ(T ) = ϕ(T )+0 = ϕ(T )+ϕ(T1) = ϕ(T ∪ T1)+ϕ(T ∩ T1) = ϕ(T2)+0 = 0. Proposition 2.5. Suppose that ϕ is a continuous invariant simple valuation on P(H2), and suppose that ϕ(I ) = 0 for all ideal triangles I ⊆ H2. Then ϕ(T ) = 0 for all triangles T ∈ P(H2). Proof. Define an invariant simple valuation ψ on closed arcs of the unit circle as follows. Suppose that Lα is a closed arc in S1 of length α, where α ∈ [0, π ). Let Sα be the semi-ideal hyperbolic triangle having non-zero angle α induced by the disc model D2 for the hyperbolic plane, where we have chosen a fixed base point x0 as the center of D2. (See Fig. 2.) Define ψ (Lα) = ϕ(Sα). The function ψ is well-defined since ϕ is invariant (and since any two semi-ideal triangles having the same non-zero angle α are congruent by some isometry). Moreover, the function ψ is a valuation on arcs of the circle, because ϕ vanishes on ideal triangles. To see this, suppose that two arcs Lα and Lβ are adjacent in the circle, having endpoints A B and BC , respectively. Then ψ (Lα) + ψ (Lβ ) = ψ ( A B) + ψ (BC ) = ϕ( AO B) + ϕ( B OC ) = ϕ( AOC ) + ϕ( A BC ) = ψ ( AC ) + 0 = ψ (Lα+β ), since A BC is ideal, as in Fig. 2. Groemer’s Theorem 1.3 then implies that ψ has a unique well-defined extension to all finite unions of arcs in S1, including arcs longer than π . We now consider the case α = π . In this instance Sπ is a line segment having area zero, so that ϕ(Sπ ) = 0 by the simplicity of ϕ. It now follows from Proposition 2.3 that ψ (Lqπ ) = 0 for all non-negative rational numbers q. This argument holds regardless of which base point in D2 or H2 we choose to play the role of the center x0, since a change of center can be accomplished by an isometry. It follows that ϕ(Sα) = 0 for any triangle Sα having two vertices at infinity and angle α = qπ at the remaining vertex in H2, where q is any rational number. Note that α K = T1 ∪ · · · ∪ Tm , ν(K ) = m i=1 ν(Ti ) = 0, ϕ(K ) = c A(K ), varies continuously as the center x0 is moved along a hyperbolic line in H2, while the two vertices at infinity remain fixed. (Although this motion is, of course, not an isometry.) Because ϕ is continuous, it now follows that ϕ vanishes on all semi-ideal triangles. By Proposition 2.4 the valuation ϕ must vanish on all triangles. Proof of Theorem 2.1. Suppose that ϕ is a continuous invariant simple valuation on P (H2). It follows from Proposition 2.2 that ϕ is finite. Let I denote an ideal triangle in H2, and let c = (1/π )ϕ(I ). Define ν(K ) = ϕ(K ) − c A(K ) for all K ∈ P (H2). Recall that an ideal triangle I in H2 has area π , so that ν(I ) = 0. Since the valuation ν satisfies the conditions of Proposition 2.5, it follows that ν(T ) = 0 for all triangles T ∈ P (H2). For a general hyperbolic polygon K ∈ P (H2) we can express K as a union of triangles where dim(Ti ∩ Tj ) < 2 for all i = j . Since ν is simple, it follows that for all K ∈ P (H2), so that for all K ∈ P (H2). Since any continuous invariant simple valuation ϕ on K(H2) restricts such a valuation on the dense subspace P (H2) of convex polytopes, the continuity of ϕ, combined with Theorem 2.1, immediately yields the following. Corollary 2.6. Suppose that ϕ is a continuous invariant simple valuation on K(H2). Let I denote an ideal triangle in H2. Then ϕ(K ) = (ϕ(I )/π ) A(K ), for all K ∈ K(H2). Invariant Valuations on the Hyperbolic Plane Theorem 2.1 can be used to characterize all isometry-invariant valuations on polygonal regions of the hyperbolic plane. To this end we must first address the one-dimensional case. It is an immediate consequence of Proposition 2.3 that a continuous simple translation-invariant valuation on compact closed intervals of R must be a multiple of Euclidean length. More generally, it follows (or see [KR]) that any continuous translation-invariant valuation on closed intervals of R must be a linear combination of Euclidean length and the Euler characteristic χ . However, in the hyperbolic plane we must also allow for polygons that have vertices at infinity. In the one-dimensional context this means we must allow for valuations that are defined on closed rays as well as closed intervals. Denote by P (H1) the collection of all finite unions and intersections of closed rays and closed intervals in the real line. Note that closed rays have infinite length and Euler characteristic zero (since a closed ray consists of one 0-cell and one 1-cell). Aside from length and the Euler characteristic χ there is an additional valuation defined on P (H1) that is continuous and translation invariant. Define the valuation χ∞ on K ∈ P (H1) by χ∞(K ) = al→im∞ χ (K ∩ {a, −a}). Since χ is a valuation, it follows that χ∞ is also a valuation. Evidently χ∞(K ) = 0 whenever K is a closed interval—indeed, whenever K is compact. Meanwhile χ∞(K ) = 1 whenever K is a closed ray, while χ∞(H1) = 2. Evidently χ∞ is also continuous and isometry invariant. Proposition 3.1. Suppose that ϕ is a continuous isometry invariant valuation on P (H1). If ϕ takes finite values on closed rays, then there exist constants c0, c∞ ∈ R such that ϕ(K ) = c0χ (K ) + c∞χ∞(K ), for all K ∈ P (H1). If ϕ takes either value ±∞ on closed rays, then there exist constants c0, c1 ∈ R such that ϕ(K ) = c0χ (K ) + c1Length(K ), for all K ∈ P (H1). Proof. Since ϕ is invariant, ϕ takes the same value on all singletons. Let c0 = ϕ({o}). Similarly, since ϕ is invariant, ϕ takes the same value on all closed rays. Suppose this is a finite value c∞ ∈ R. Write R as a union of two rays R1, R2 (positive and negative) sharing a common endpoint at the origin, {o} = R1 ∩ R2. Then ϕ(R) = ϕ( R1) + ϕ( R2) − ϕ( R1 ∩ R2) = 2c∞ − c0. More generally, if C is any closed interval, we can express C is an intersection of two rays R1, R2 whose union is all of R, so that ϕ(C ) = ϕ( R1) + ϕ( R2) − ϕ(R) = c∞ + c∞ − (2c∞ − c0) = c0. It follows that ϕ(K ) = c0χ (K ) + c∞χ∞(K ) for all K ∈ P (H1) This completes the proof for the case in which ϕ takes a finite value on a closed ray. Suppose instead that ϕ takes either value ±∞ on closed rays. Once again, since ϕ is invariant, ϕ takes the same value c0 on all singletons. Since the valuation ϕ − c0χ now vanishes on singletons (points), it follows from Proposition 2.3 that ϕ − c0χ is a constant multiple of length when applied to closed intervals. In other words, there exists c1 ∈ R such that ϕ = c0χ + c1Length when applied to finite unions of points and closed intervals. The valuation χ∞ is extended to polygons in H2 as follows. Choose a base point x0 ∈ H2 and let Cr denote the set of points that lie at a distance r > 0 from x0. For convex polygons K ∈ P (H2) define χ∞(K ) = rl→im∞ χ (K ∩ Cr ). ( 1 ) Since χ is a valuation, it follows that χ∞ is also a valuation. Evidently χ∞(K ) = 0 whenever K is compact, since K ∩ Cr = ∅ for sufficiently large r when K is compact. More generally, for a convex polygon K the value of χ∞(K ) is exactly the number of “vertices at infinity” of K . For example, χ∞ = 3 for ideal triangles, while χ∞ = 1 for rays and χ∞ = 2 for hyperbolic lines, generalizing the definition above for the case of H1. Evidently χ∞ is independent of the choice of base point x0. Moreover, for all K ∈ K(H2) we have χ (K ) + χ∞(K ) = 1. Recall that the length functional on the hyperbolic line H1 extends to the hyperbolic perimeter functional 12 P on polygons in H2, where the normalization factor of 12 makes the perimeter functional continuous—a line segment in H2 is a hyperbolic “2-gon”, whose perimeter is twice its hyperbolic length, since the line segment can be expressed as the limit of a sequence of flattening triangles. Theorem 2.1 can now be applied to derive following characterization theorem for continuous invariant valuations on polygons and convex bodies in H2. Theorem 3.2 (Invariant Valuation Characterization Theorem for H2). Suppose that ϕ is a continuous invariant valuation on P(H2). If ϕ takes finite values on closed rays, then there exist c0, c2, c∞ ∈ R such that ϕ(K ) = c0χ (K ) + c2 A(K ) + c∞χ∞(K ), for all K ∈ P(H2). If ϕ takes either value ±∞ on closed rays, then there exist c0, c1, c2 ∈ R such that ϕ(K ) = c0χ (K ) + c1 P(K ) + c2 A(K ), for all K ∈ P(H2). Since the set P(H2) is dense in K(H2), Theorem 3.2 also holds if P(H2) is replaced with the larger collection K(H2). Theorem 3.2 provides a partial analogue of Hadwiger’s Characterization Theorem 0.1, as described in the Introduction (see also [Ha], [Kl1], and [KR]). Note that Theorem 3.2 may seem incomplete, since it does not appear to account for the valuation P + χ∞, for example. However, this is not a problem, since P + χ∞ = P. Since χ∞ vanishes on all compact polygonal regions, we can add any scalar multiple of χ∞ to a valuation having a non-trivial P component without changing the valuation on any K ∈ K(H2). Proof of Theorem 3.2. Let ϕ denote a continuous invariant valuation on P(H2). Suppose that ϕ takes finite values on closed rays. By Proposition 3.1 the restriction of ϕ to a hyperbolic line has the form ϕ = c0χ + c∞χ∞, where c0, c∞ ∈ R are constants independent of the choice of hyperbolic line (because ϕ is isometry invariant). It follows that the valuation ν on P(H2) given by ν = ϕ − c0χ − c∞χ∞ vanishes on all K ∈ P(H2) of dimension less than 2; that is, ν is a continuous invariant simple valuation on P(H2). Theorem 2.1 then implies the existence of c2 ∈ R such that ν(K ) = c2 A(K ) for all K ∈ P(H2). Suppose instead that ϕ takes infinite values on closed rays. A similar argument using Proposition 3.1 yields the analogous result, in which χ∞ is replaced by the perimeter P. Note that valuations of type ( 1 ) in Theorem 3.2 are constant on line segments. This provides a simple test for when χ∞ can be omitted from consideration. Corollary 3.3. Suppose that ϕ is a continuous invariant valuation on P(H2) that is not constant on line segments. Then there exist c0, c1, c2 ∈ R such that, for all K ∈ P(H2), ϕ(K ) = c0χ (K ) + c1 P(K ) + c2 A(K ). Meanwhile, a finiteness condition will determine when the perimeter P is omitted. Corollary 3.4. Suppose that ϕ is a continuous invariant finite valuation on P(H2). Then there exist c0, c2, c∞ ∈ R such that, for all K ∈ P(H2), ϕ(K ) = c0χ (K ) + c2 A(K ) + c∞χ∞. Theorem 3.2 also implies that any continuous invariant valuation on P(H2) is determined up to a multiple of χ∞ by its values on a hyperbolic disc Dr of radius r . Recall that P(Dr ) = 2π sinh r and A(Dr ) = 2π(cosh r − 1). ( 2 ) See, for example, p. 85 of [St]. Since χ (Dr ) = 1 for all r ≥ 0, we have ϕ(Dr ) = c0 + 2π c1 sinh r + 2π c2(cosh r − 1). The coefficients ci in Theorem 3.2 are easily computed once ϕ(Dr ) is known for three suitable values of r . 4. Integral Geometry in the Hyperbolic Plane Hadwiger’s Characterization Theorem 0.1 for invariant valuations on Euclidean space provides a powerful mechanism for deriving fundamental integral-geometric identities. For a number of applications and consequences of Hadwiger’s theorem, see, for example, [Ha] and [KR]. The Area Theorem 2.1 and the equivalent characterization Theorem 3.2 provide similar advantages in the context of hyperbolic integral geometry. A simple though fundamental example is the area formula for hyperbolic triangles and polygons, a special case of the Gauss–Bonnet theorem [St, p. 100]. Corollary 4.1 (Gauss–Bonnet Theorem for Polygons). Suppose that K is a simple closed polygonal curve in H2, and suppose that the boundary of K has n vertices (possibly at infinity), with corresponding interior angle measures α1, . . . , αn ∈ [0, π ]. Then the area of K is given by A(K ) = (n − 2)π(χ (K ) + χ∞(K )) − n i=1 αi . Note that the χ∞ term vanishes when K is compact. Proof. For a convex polygon K define (K ) to be the sum of the angles between unit outer normals of K wherever two adjacent edges meet at a vertex. If K and L are convex polygons such that K ∪ L is also convex, then the boundaries ∂ K and ∂ L must meet either at vertices or at edges having the same unit normal, so that (K ∪ L) + (K ∩ L) = (K ) + (L). It follows from Groemer’s Theorem 1.3 that extends to a valuation on P(H2). Evidently is invariant and continuous. Moreover, = 2π for all points, line segments, and rays. By Theorem 3.2 there exist a, b, c ∈ R such that = aχ +b A+cχ∞. Since = 2π on points, a = 2π . Since = 2π on rays, c = 2π . Because = 3π on ideal triangles (while χ + χ∞ = 1 and A = π ) we obtain = 2π(χ + χ∞) + A. Let σ (K ) denote the sum of the interior angles at the vertices of K . (Note that σ is not a valuation.) If K has n vertices then σ (K ) + (K ) = π n, so that A(K ) = (K )−2π(χ (K )+χ∞(K )) = (π n −σ (K ))−2π(χ (K )+χ∞(K )) = · · · = (n − 2)π(χ (K ) + χ∞(K )) − αi . n i=1 Recall that if T is a hyperbolic triangle then n = 3 and χ (T ) + χ∞(T ) = 1. Corollary 4.2 (Area of a Hyperbolic Triangle). If a triangle T in H2 has interior angle measures α, β, γ ∈ [0, π ] then the area of T is then given by the angle deficit: A(T ) = π − (α + β + γ ). Theorem 3.2 also yields a quick proof of the principal kinematic formula for compact convex sets in H2, a fundamental theorem of geometric probability [KR], [San]. While a classical proof of the principal kinematic formula can be found in [San, p. 321], Theorem 3.2 immediately implies the following stronger result. Theorem 4.3 (Kinematic Formula for Invariant Valuations on H2). Suppose that ϕ is a continuous isometry invariant valuation on P(H2) (or K(H2)). Then there exists a constant real 4 × 4 symmetric matrix C such that g ϕ(K ∩ g L) dg = χ (K ) P(K ) The integral on the left-hand side of ( 3 ) is taken with respect to the hyperbolic area on H2 and the invariant Haar probability measure on the group G0 of hyperbolic isometries that fix a base point x0 ∈ H2. To define this more precisely, denote by tx the unique hyperbolic translation of H2 that maps x0 to a point x ∈ H2. Then define g where we use the probabilistic normalization g∈G0 dg = 1. ( 3 ) (4) Proof of Theorem 4.3. To begin, define ϕ(K , L) = ϕ(K ∩ g L) dg. g For fixed K , the set function ϕ(K , L) is a valuation in the variable L; in fact, it is an invariant valuation, since ϕ(K , g0 L) = ϕ(K ∩ gg0 L) dg = ϕ(K ∩ g L) dg, for each isometry g0. It follows from Theorem 3.2 that we can express ϕ(K , L) as a linear combination of the valuations χ , P, A, χ∞, with coefficients ci (K ) depending on K : ϕ(K , g0 L) = c0(K )χ (L) + c1(K ) P(L) + c2(K ) A(L) + c∞(K )χ∞(L). Meanwhile, for fixed L, the set function ϕ(K , L) is a valuation in the variable K . It follows that each of the coefficients ci (K ) is a valuation in the variable K . Moreover, since the valuation ϕ and the Haar integral in the left-hand side of ( 3 ) are both isometry invariant, we have ϕ(K , L) = Therefore, each ci (K ) is an invariant valuation in the variable K , so that Theorem 3.2 applies, yielding the matrix equation ϕ(K , L) = χ (K ) P(K ) where C = [ci j ]i, j∈{0,1,2,∞} is a 4 × 4 matrix of real constants, independent of K and L. Since ϕ(K , L) = ϕ(L , K ), it follows that ci j = cji . The following special case is of fundamental importance in integral geometry and geometric probability. See, for example, [Fu], [Ho], [KR], [San], and [SW]. Corollary 4.4 (Principal Kinematic Formula for H2). For K , L ∈ K(H2), g 1 1 χ (K ∩ g L) dg = χ (K ) A(L)+ 2π P(K ) P(L)+ A(K )χ (L)+ 2π A(K ) A(L). (5) In order to verify (5) we require the notion of the parallel body. The parallel body of K having radius ε ≥ 0 is the set Kε of points in H2 (or Hn) whose (hyperbolic) Hausdorff distance to the set K is at most ε. See, for example, [Sc1]. Let Dε denote the set of points that lie at most a distance ε from x0, where x0 is a chosen base point. Note that g Dε = Dε for all isometries g ∈ G0. When K is a compact convex set, the indicator function of Kε is given by IKε (x ) = χ (K ∩ tx (Dε)) = Since χ is a valuation and integration is linear, it follows that the mapping K → A(Kε) is a valuation in the parameter K (where ε is a fixed constant). Proof of Corollary 4.4. In order to compute the values of the coefficients ci j , we evaluate χ (K , L) by calculating for the cases in which K = L = Dr , for some r ≥ 0. For example, it is evident from (4) that χ (K , L) = 0 when K and L are points, or when K is a point and L is a line segment, so that c00 = c10 = c01 = 0. More generally, if L is a point, then χ (K , L) = χ (K , D0) = A(K ) = A(K )χ (L) by (6), so that c0 j = cj0 = 0 for all j = 2, while c02 = c20 = 1. Next, note that if Ka is a line segment of length a and dim L ≥ 1, then the family of motions of L that meet Ka is strictly increasing as a increases. By Corollary 3.3 the valuation χ∞ does not appear anywhere in the expression for χ (K , L), so that c∞ j = cj∞ = 0 for all j . To compute the remaining ci j we use the identity P(Da)2 = π A(D2a) for a ≥ 0, (7) an elementary consequence of ( 2 ). Denote by ∂ D the boundary of a hyperbolic disk D. Note that χ (∂ D) = A(∂ D) = 0, while P(∂ D) = 2 P(D), since the “perimeter” of a one-dimensional curve is twice its length. (Recall that a one-dimensional curve is, in the limiting sense, a “two-sided” polygon.) We now compute χ (∂ Da, ∂ Da), χ (∂ Da, Da), and χ (Da, Da). Since Da is the set of points which lie at most a distance a from x0, we have g Da = Da for all isometries g ∈ G0, and similarly for ∂ Da. By (6), χ (K , Da) = A(Ka), where Ka is the a-parallel body of K . In particular, χ (Da, Da) = A(D2a). If two closed disks intersect, they do so as a compact set, while their boundaries generically intersect in exactly two points. In particular, since χ (Da ∩ tx (∂ Da)) = χ (Da ∩ tx (Da)) for all x = x0, while χ (∂ Da ∩ tx (∂ Da)) = 2χ (Da ∩ tx (Da)) for all x ∈/ ∂ D2a with x = x0. Therefore 1 12 χ (∂ Da, ∂ Da) = χ (Da, Da) = A(D2a) = π P(Da)2, (8) by (7). Since χ (∂ Da) = A(∂ Da) = 0, it follows from ( 3 ) and (8) that 2 π P(Da)2 = χ (∂ Da, ∂ Da) = c11 P(∂ Da) P(∂ Da) = 4c11 P(Da)2, so that c11 = 1/2π . Similar elementary considerations lead to c22 = 1/2π and c12 = c21 = 0, completing the proof of the kinematic formula (5). The following corollary can be derived directly (see p. 322 in [San]), but follows immediately from (6) and Corollary 4.4. Corollary 4.5 (The Area of a Parallel Body). For K ∈ K(H2) and ε ≥ 0, 1 1 + 2π A(Dε) A(K ) = 2π(cosh ε − 1) + (sinh ε) P(K ) + (cosh ε) A(K ). Corollary 4.5 gives the hyperbolic analogue of Steiner’s formula for the area (or volume) of a Euclidean parallel body [Sc1]. It would be interesting to see how a suitable variation of Corollary 4.4 (possibly using integration over a suitable chosen proper subgroup of isometries) might yield hyperbolic analogues of Minkowski’s mixed volumes and the related Brunn–Minkowski theory [Sc1]. Corollary 4.4 and its higher-dimensional generalizations have numerous applications to questions in geometric probability, leading, for example, to hyperbolic analogues of Hadwiger’s containment theorem for planar regions [KR], [San] and to Bonnesen’s inequality for area (see [Kl3] and [San, p. 120]), a generalization of the classical isoperimetric inequality (see also [Os]). Variations of these kinematic techniques can also be found in [Kl3] and [San, p. 324]. 5. Characterizing Valuations in Hn The proof of the hyperbolic area characterization, Theorem 2.1, relied in part on a relationship between an invariant valuation on H2 and a derived invariant valuation on the unit circle, which was in turn easily characterized. Equally important was the fact that all ideal triangles in H2 are congruent with respect to an hyperbolic isometry. For dimensions n ≥ 3, ideal simplices in Hn are no longer necessarily congruent. Moreover, while spherical area in the two-dimensional sphere S2 has a valuation characterization (see p. 156 in [KR]), the analogous characterization of spherical volume in Sn remains an open conjecture for n ≥ 3. As a result, the methods of the previous sections do not entirely generalize to higher-dimensional hyperbolic space. In order to extend some of the previous results to higher-dimensional hyperbolic space, we make do with the following partial result regarding invariant valuations on spherical polytopes in an even-dimensional sphere S2n. Define a lune in S2n to be a subset of S2n consisting of the intersection of at most 2n hemispheres. Theorem 5.1. Suppose that ϕ is an isometry-invariant simple valuation on P(S2n). If ϕ(L) = 0 for all lunes L ⊆ S2n, then ϕ(K ) = 0 for all K ∈ P(S2n). Theorem 5.1 will play a role analogous to that of Proposition 2.3 characterizing hyperbolic volume in higher dimensions. Note that continuity plays no role in this theorem. Theorem 5.1 can be found on p. 165 in [KR]. For completeness we include a proof here. Proof of Theorem 5.1. Suppose that is a spherical simplex in S2n given by the intersection of hemispheres . For X ⊆ S2n, denote by=X Hc1th∩e c·l·o·s∩urHe2onf+t1he complement S2n − X . Note that 2n+1 i=1 Hi c = 2n+1 i=1 H c i = − . Because ϕ vanishes on lunes, ϕ(S2n) = 0. Since ϕ is simple and invariant, Hi = ϕ(S2n) − ϕ = 0 − ϕ(− ) = −ϕ( ). (9) ϕ( ) = ϕ Hi = −ϕ( ), (11) so that ϕ( ) = 0. Since every polytope K ∈ P(S2n) can be expressed as a union of spherical simplices intersecting in dimension less than 2n, it follows that ϕ(K ) = 0 for all K ∈ P(S2n). The sign discrepancy in (11), which in turn implies that ϕ = 0 identically, depends on the fact that the inclusion–exclusion expansion on the right-hand side of (10) terminates ϕ 2n+1 i=1 2n+1 i=1 Hi c 2n+1 i=1 Meanwhile, since ϕ is a valuation, the inclusion–exclusion identity yields ϕ 2n+1 i=1 2n+1 i=1 i1<i2 Hi = ϕ(Hi )− ϕ(Hi1 ∩ Hi2 )+· · ·+(−1)2nϕ(H1 ∩· · ·∩ H2n+1). (10) Since ϕ vanishes on hemispheres and lunes, all of the terms on the right-hand side of (10) vanish except possibly for the last term: (−1)2nϕ(H1 ∩ · · · ∩ H2n+1) = ϕ( ). Combining (9) and (10) then yields with an even power of −1, a consequence of the even-dimensionality of S2n. A version of Theorem 5.1 for S2n+1 remains an open problem. An hyperbolic n-simplex S is called semi-ideal if at least n of its n + 1 vertices lie on the plane at infinity. Proposition 5.2. Suppose that ϕ is an invariant simple valuation on P(Hn), and suppose that ϕ(S) = 0 for all ideal and semi-ideal simplices S ⊆ Hn. Then ϕ(K ) = 0 for all polytopes K ∈ P(Hn). Proof. Suppose T is a simplex in Hn having at least two vertices x0 = x1 that do not lie at infinity. Let x ∗ denote the point at infinity that is collinear with x0 and x1, so that x1 lies between x0 and x ∗. If T = [x0, x1, x2, . . . , xn], let T1 and T2 denote the simplices defined by T1 = [x ∗, x1, x2, . . . , xn] and T2 = [x ∗, x0, x2, . . . , xn]. Then T ∪ T1 = T2, while dim(T ∩ T1) ≤ n − 1. Since ϕ is a simple valuation, we have ϕ(T ) + ϕ(T1) = ϕ(T ∪ T1) + ϕ(T ∩ T1) = ϕ(T2) + 0 = ϕ(T2). (12) We now proceed by induction on the number of vertices at infinity. If T has n − 1 or more vertices at infinity, then T is semi-ideal, so that ϕ(T ) = 0, by our initial assumption. Suppose that ϕ(T ) = 0 whenever T has k vertices at infinity, for some k ≥ 1. If a simplex T has k − 1 vertices at infinity, then (12) implies that ϕ(T ) = ϕ(T2) − ϕ(T1), for some T1, T2 having k vertices at infinity. Hence, ϕ(T ) = 0 as well. It follows that ϕ(T ) = 0 for all n-simplices T . Since every polytope K ∈ P(Hn) has a finite simplicial decomposition, ϕ vanishes on P(Hn). For a compact convex set K in Hn, denote by Vn(K ) the hyperbolic volume of K . Theorem 5.3 (Ideal Determination Theorem—Finite Case). Suppose that ψ1 and ψ2 are invariant finite simple valuations on P(H2n+1), where n is a positive integer. If ψ1(I ) = ψ2(I ) for all ideal (2n + 1)-dimensional simplices I in H2n+1, then ψ1(K ) = ψ2(K ), for all K ∈ P(H2n+1). Note: We do not assume that the valuations ψi in Theorem 5.3 are continuous. Proof. Let ϕ = ψ1 − ψ2, so that ϕ(I ) = 0 for all ideal I . It now suffices to show that ϕ is identically zero on all polytopes. Define an invariant simple valuation η on convex spherical polytopes in S2n as follows. Choose a base point x0 ∈ H2n+1. Suppose that L is a convex spherical polytope in S2n having extreme points z1, . . . , zm that all lie in the same open hemisphere of S2n. Let Q L denote hyperbolic convex hull of the points x0, z1, . . . , zm in H2n+1, where S2n is viewed as the boundary of the Poincare´ ball model D2n+1 for the hyperbolic space H2n+1 having center at x0. Define η(L) = ϕ(Q L ). The function η is an orthogonal invariant since ϕ is invariant (while isometries of S2n are restrictions of certain hyperbolic isometries of H2n+1). Moreover, the function η satisfies the inclusion–exclusion condition for valuations, because ϕ vanishes on ideal simplices. (See, for example, Fig. 2.) Groemer’s Theorem 1.3 then implies that η has a unique well-defined extension to all finite unions of convex spherical polytopes, including those no longer contained in an open hemisphere. The simplicity of η also follows immediately from the simplicity of ϕ. If L is a lune in S2n consisting of the intersection of exactly 2n generically positioned hemispheres, then we call L a minimal lune. In this case L contains exactly one pair of antipodal points a, a ∈ S2n, and L = L1 ∪ L2, where L1 and L2 are spherical simplices congruent by a reflection of S2n across the great-(2n − 1)-subsphere Z normal to the axis aa . Note that Q L1 (resp. Q L2 ) is a semi-ideal simplex having one vertex at a (resp. a ), one vertex at x0, and the remaining vertices in the great subsphere Z . Since a and a are antipodal points, the union Q L1 ∪ Q L2 is an ideal simplex having a vertex at each of a and a and the remaining vertices in Z . Since L1 ∩ L2 has dimension less than 2n in S2n, we have η(L1 ∩ L2) = 0. Similarly, since Q L1 ∩ Q L2 has dimension less than 2n + 1, we have ϕ(Q L1 ∩ Q L2 ) = 0. It follows that η(L) = η(L1) + η(L2) = ϕ(Q L1 ) + ϕ(Q L2 ) = ϕ(Q L1 ∪ Q L2 ) + 0 = 0, since ϕ vanishes on ideal simplices. If L is a lune consisting of an intersection of fewer than 2n hemispheres, then L can be subdivided into a finite union of minimal lunes, so that η(L) = 0 once again, by the simplicity property of η. Theorem 5.1 then implies that η(K ) = 0 for all K ∈ P(S2n). In particular, ϕ vanishes on all hyperbolic simplices having one vertex at x0 (or, since our choice of x0 was arbitrary, at any other point of H2n+1) and all remaining vertices at infinity (i.e., on S2n, using the Poincare´ ball model for H2n+1). In other words, ϕ vanishes on all semiideal simplices in H2n+1. It follows from Proposition 5.2 that ϕ must vanish on all of P(H2n+1). The proof of the following proposition is exactly analogous to the proof of Proposition 2.2. Proposition 5.4. Suppose that ϕ is a continuous invariant simple valuation on P(Hn). Then ϕ is finite on P(Hn). We are now ready to state and prove a partial generalization of the Area Theorem 2.1. Theorem 5.5 (Ideal Determination Theorem—Continuous Case). Suppose that ψ1 and ψ2 are continuous invariant simple valuations on P(H2n+1), where n is a positive integer. If ψ1(I ) = ψ2(I ) for all ideal (2n + 1)-dimensional simplices I in H2n+1, then ψ1(K ) = ψ2(K ), for all K ∈ P(H2n+1). Proof. Since each of the valuations ψi is continuous, invariant, and simple on P(H2n+1), each ψi is also a finite valuation by Proposition 5.4, so that Theorem 5.3 applies, and ψ1(K ) = ψ2(K ) for all K ∈ P(H2n+1). Note: If each of the valuations ψi in Theorem 5.5 is defined on all of K(H2n+1), then the continuity of the ψi and the density of P(H2n+1) in K(H2n+1) together imply that ψ1(K ) = ψ2(K ) for all K ∈ K(H2n+1) as well. In other words, continuous invariant simple valuations on H2n+1 are completely determined by their values on ideal simplices. In the case of dimension 3, we can also remove the condition of simplicity. Corollary 5.6 (Ideal Determination Theorem for H3). Suppose that ψ1 and ψ2 are invariant continuous valuations on P(H3). If ψ1 and ψ2 agree on singletons (points), and if (ψ1 − ψ2)(I ) = 0 for all ideal simplices I in H3, then (ψ1 − ψ2)(K ) = 0, for all K ∈ K(H3). Evidently the conclusion of Corollary 5.6 implies that ψ1 = ψ2 as valuations on K(H3). In the hypothesis we require (ψ1 − ψ2)(I ) = 0 rather than ψ1(I ) = ψ2(I ) in order to account more carefully for the possibility that ψ1 and ψ2 take infinite values on some ideal simplex. Proof. Let ξ denote a two-dimensional hyperbolic plane inside H3. Since the difference ψ1 − ψ2 vanishes on hyperbolic lines (ideal 1-simplices), it follows from Theorem 3.2 that (ψ1 − ψ2)|ξ = c0χ + c χ ∞ ∞ + c2 A, where A denotes the hyperbolic area in ξ . Since ψ1 − ψ2 vanishes on points, it follows that c0 = 0. Since ψ1 − ψ2 vanishes on hyperbolic lines, c∞ = 0 as well. If I is an ideal triangle inside ξ then 0 = (ψ1 − ψ2)(I ) = c2 A(I ) = c2π, so that c2 = 0. In other words, the invariant valuation ψ1 − ψ2 vanishes on all polygons in ξ , and therefore in any two-dimensional flat of H3, so that ψ1 − ψ2 is simple. Theorem 5.5 now applies to the simple valuation ψ1 − ψ2. Since (ψ1 − ψ2)(I ) = 0 for all three-dimensional ideal simplices in H3, it follows that (ψ1 − ψ2)(K ) = 0 for all polytopes K , and, by continuity, for all K ∈ K(H3). Open Problem. Theorems 5.3 and 5.5 are stated only for odd-dimensional hyperbolic spaces. The unsolved case for even-dimensional hyperbolic spaces is a gap that hinders the generalization of Corollary 5.6 to dimension 4 or greater. This limitation stems from Theorem 5.1, which has only been proven for even-dimensional spherical spaces. 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Daniel A. Klain. Isometry-Invariant Valuations on Hyperbolic Space, Discrete & Computational Geometry, 2006, 457-477, DOI: 10.1007/s00454-006-1251-6