Valuations on Convex Sets of Oriented Hyperplanes

Gates, John M.; Hug, Daniel; Schneider, Rolf

doi:10.1007/s00454-004-1126-2

Cited by 3 publications

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“…In recent years there have been many new developments in the theory of valuations (see [1,7,12,13,19,32,34,68,69,82,84]). Here we confine our attention to valuations in the affine geometry of convex bodies.…”

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Valuations in the Affine Geometry of Convex Bodies

Ludwig¹

2006

Integral Geometry and Convexity

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A survey on SL(n) invariant and SL(n) covariant valuations is given. 1 Introduction A function Φ defined on convex bodies in R n and taking values in an Abelian semigroup is called a valuation if Φ(K) + Φ(L) = Φ(K ∪ L) + Φ(K ∩ L) for K, L, K ∪ L ∈ K n , where K n is the set of convex bodies (convex, compact sets) in R n. Thus the notion of valuation is a generalization of the notion of measure. In the 1930s, Blaschke obtained the first classification of real valued valuations that are SL(n) invariant. This was greatly extended by Hadwiger in his famous classification of continuous and rigid motion invariant valuations.

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Valuations in the Affine Geometry of Convex Bodies

Ludwig¹

2006

Integral Geometry and Convexity

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“…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]). …”

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confidence: 99%

Isometry-Invariant Valuations on Hyperbolic Space

Klain

2006

Discrete Comput Geom

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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 *

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