Hostname: page-component-74656f9884-j7cmm Total loading time: 0 Render date: 2026-11-07T18:17:11.717Z Has data issue: false hasContentIssue false
  • English
  • Français

A short proof of Hadwiger's characterization theorem

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Daniel A. Klain
Daniel A. Klain
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, U.S.A.
Get access

Abstract

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].

MSC classification

Secondary: 52A39: Mixed volumes and related topics

Information

Type
Research Article
Information
Mathematika , Volume 42 , Issue 2 , December 1995 , pp. 329 - 339
Copyright
Copyright © University College London 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1.Chen, B. and Rota, G. C.. Totally invariant set functions of polynomial type. Communications in Pure and Applied Mathematics, 47 (1994), 187–197.CrossRefGoogle Scholar
2.Gardner, R. J.. Geometric Tomography (Cambridge University Press, 1995).Google Scholar
3.Groemer, H.. On the extension of additive functional on classes of convex sets. Pacific J. Math., 75 (1978), 397–410.CrossRefGoogle Scholar
4.Hadwiger, H.. Vorlesungen über Inhalt, Oberflàche, und Isoperimetrie (Berlin: Springer Verlag, 1957).CrossRefGoogle Scholar
5.McMullen, P.. Non-linear angle-sum relations for polyhedral cones and polytopes. Math. Proc. Comb. Phil. Soc, 78 (1975), 247–261.CrossRefGoogle Scholar
6.McMullen, P. and Schneider, R.. Valuations on convex bodies. In Convexity and Its Applications, edited by Gruber, Peter M. and Wills, Jörg M. (Boston: Birkhàuser Verlag, 1983).Google Scholar
7.Rota, G. C.. Introduction to Geometric Probability. Lezioni Lincee held at the Scuola Normale Superiore Pisa, Dec. 2-22, 1986.Google Scholar
8.Sah, C. H.. Hilbert's Third Problem: Scissors Congruence (San Francisco: Fearon Pitman Publishers Inc., 1979).Google Scholar
9.Santalo, L. A.. Integral Geometry and Geometric Probability (Reading, MA: Addison-Wesley, 1976).Google Scholar
10.Schneider, R.. Convex Bodies: The Brunn-Minkowski Theory (Cambridge University Press, 1993).CrossRefGoogle Scholar