A Simplified Elementary Proof of Hadwiger's Volume Theorem

Abstract

One of the most important results in geometric convexity is Hadwiger's characterization of quermassintergrals and intrinsic volumes. The importance lies in that Hadwiger's theorem provides straightforward proofs of numerous results in integral geometry such as the kinematic formulas [Santaló, L. A.: Integral Geometry and Geometric Probability, Addison-Wesley, 1976], the mean projection formulas for convex bodies [Schneider, R.: Convex Bodies: The Brunn—Minkowski Theory, Cambridge Univ. Press, 1993], and the characterization of totally invariant set functions of polynomial type [Chen, B. and Rota, G.-C.: Totally invariant set functions of polynomial type, Comm. Pure Appl. Math. 47 (1994), 187–197]. For a long time the only known proof of Hadwiger's theorem was his original one [Hadwiger, H.: Vorlesungen über Inhalt, Oberfläche and Isoperimetrie, Springer, Berlin, 1957] (long and not available in English), until a new proof was obtained by Klain [Klain, D. A.: A short proof of Hadwiger's characterization theorem, Mathematika 42 (1995), 329–339., Klain, D. A. and Rota, G.-C.: Introduction to Geometric Probability, Lezioni Lincee, Cambridge Univ. Press, 1997], using a result from spherical harmonics. The present paper provides a simplified and self-contained proof of Hadwiger's theorem.