Abstract
Hyperbolic area is characterized as the unique continuous isometry-invariant simple valuation on convex polygons in \({\Bbb H}^2.\) We then show that continuous isometry-invariant simple valuations on polytopes in \({\Bbb H}^{2n+1}\) for \(n \geq 1\) 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 \({\Bbb H}^2,\) a partial characterization in \({\Bbb H}^3,\) and a mechanism for deriving many fundamental theorems of hyperbolic integral geometry, including kinematic formulas, containment theorems, and isoperimetric and Bonnesen-type inequalities.
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