Isometry-Invariant Valuations on Hyperbolic Space

Hyperbolic area is characterized as the unique continuous isometry invariant simple valuation on convex polygons in H. We then show that continuous isometry invariant simple valuations on polytopes in H 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 H, a partial characterization inH, and a mechanism for deriving many fundamental theorems of hyperbolic integral geometry, including kinematic formulas, containment theorems, and isoperimetric and Bonnesen-type inequalities. 2000 AMS subject classification: 52A55; 52A38; 52B45. A valuation on polytopes, convex bodies, or more general class of sets, is a finitelyadditive 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) [KR97, McM93]. Other important valuations on convex bodies and polytopes include projection functions and cross-section measures [Gar95, KR97, Sch93], affine surface area [Lut93, Lut91], and Dehn invariants [Sah79]. 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 [Rud87], valuations form a more general class of set functionals that has so far resisted such sweeping classifications [KR97, McM93]. 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 Euclidean space. Such characterizations have had fundamental impact in convex, integral, and combinatorial geometry [Ale99, Ale00, Ale01, Had57, KR97, Kla00, Kla04, Lud99, LR99, Sch96, McM77, McM89, McM93] as well as to the theory of dissection of polytopes [Bol79, Had57, KR97, McM93, Sah79]. 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 [McM93]): Research supported in part by NSF grant #DMS-9803571.