Isometry-Invariant Valuations on Hyperbolic Space

Abstract. Hyperbolic area is characterized as the unique continuous isometry invariant simple valuation on convex polygons in H2 . We then show that continuous isometry invariant 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. 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 ...

• Type: Journal
• Year: 2006
• Where: DCG
• Authors: Daniel A. Klain
• Keywords: DCG 2006, Invariant Simple Valuations, Isometry Invariant Valuations, Polytopes