Research Article

A short proof of Hadwiger's characterization theorem

Daniel A. Klaina1

a1 School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, U.S.A.


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].

(Received September 30 1994)

Key Words:

  • 52A39: CONVEX AND DISCRETE GEOMETRY; General Convexity; Mixed volumes and related topics.