Isometry-Invariant Valuations on Hyperbolic Space

Abstract

Abstract. 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 H 2n+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 H 2 , a partial characterization in H 3 , and a mechanism for deriving many fundamental theorems of hyperbolic integral geometry, including kinematic formulas, containment theorems, and isoperimetric and Bonnesen-type inequalities. IntroductionA 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. [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 *

Bonnesen-type inequalities for surfaces of constant curvature

“…The classical proof of Theorem 1.1 can be found in [San76]. For a valuation-based proof of Theorem 1.1, see [KR97] (for the Euclidean and spherical cases) and [Kla03] (for the hyperbolic plane). Surveys and other recent work on kinematic formulas in convex, integral, and Riemannian geometry and their applications include [Fu90, How93, KR97, San76, SW93, Zha99].…”