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A short proof of Hadwiger’s characterization theorem. (English) Zbl 0835.52010
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). Hadwider’s characterization leads to effortless proofs of numerous results in integral geometry, including various kinematic formulas and the mean projection formulas for convex bodies, and also provides a connection between rigid motion invariant set functions and symmetric polynomials. The purpose of this paper is to present a new and short proof of Hadwiger’s characterization theorem. En route to this result we give a more general characterization of volume in Euclidean space.

52A39 Mixed volumes and related topics in convex geometry
52A22 Random convex sets and integral geometry (aspects of convex geometry)
Full Text: DOI
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