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A short proof of Hadwiger's characterization theorem

  • Daniel A. Klain (a1)
  • DOI:
  • Published online: 01 February 2010

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].

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1.B. Chen and G. C. Rota . Totally invariant set functions of polynomial type. Communications in Pure and Applied Mathematics, 47 (1994), 187197.

3.H. Groemer . On the extension of additive functional on classes of convex sets. Pacific J. Math., 75 (1978), 397410.

4.H. Hadwiger . Vorlesungen über Inhalt, Oberflàche, und Isoperimetrie (Berlin: Springer Verlag, 1957).

6.P. McMullen and R. Schneider . Valuations on convex bodies. In Convexity and Its Applications, edited by Peter M. Gruber and Jörg M. Wills (Boston: Birkhàuser Verlag, 1983).

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