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A short proof of Hadwiger's characterization theorem

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Daniel A. Klain
Daniel A. Klain
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, U.S.A.
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Abstract

One of the most beautiful and important results in geometric convexity is Hadwiger's characterization theorem for the quermassintegrals. Hadwiger's theorem classifies all continuous rigid motion invariant valuations on convex bodies as consisting of the linear span of the quermassintegrals (or, equivalently, of the intrinsic volumes) [4]. Hadwiger's characterization leads to effortless proofs of numerous results in integral geometry, including various kinematic formulas [7, 9] and the mean projection formulas for convex bodies [10]. Hadwiger's result also provides a connection between rigid motion invariant set functions and symmetric polynomials [1, 7].

MSC classification

Secondary: 52A39: Mixed volumes and related topics
Type
Research Article
Information
Mathematika , Volume 42 , Issue 2 , December 1995 , pp. 329 - 339
Copyright
Copyright © University College London 1995

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References

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