A short proof of Hadwiger's characterization theorem

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]. Unfortunately the only known proof of Hadwiger's result until now has been that given in [4] and is the product of a long and arduous sequence of cut and paste arguments. 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. The proof relies almost entirely on elementary techniques, with the exception of Proposition 3.1, a well-known consequence of the theory of spherical harmonics.