A short proof of Hadwiger's characterization theorem,
Mathematika, 42 1995, 329-339.

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). Hadwiger's characterization leads to effortless proofs of numerous results in integral geometry, including various kinematic formulas and mean projection formulas for convex bodies. Hadwiger's result also provides a connection between rigid motion invariant set functions and symmetric polynomials.

The purpose of this paper is to present a new and shorter proof of Hadwiger's characterization theorem, digestible within a few minutes.

En route to this result is a more general characterization of volume in Euclidean space, as the unique continuous translation invariant simple even valuation on compact convex sets.

A pdf file for this paper is available here.

Characterization of even valuations is treated in the sequel, Even valuations on convex bodies.

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