A classification of $\operatorname{SL}(n)$ invariant valuations

Monika Ludwig; Matthias Reitzner

doi:http://doi.org/10.4007/annals.2010.172.1219

Pages 1219-1267 from Volume 172 (2010), Issue 2 by Monika Ludwig, Matthias Reitzner

Abstract

A classification of upper semicontinuous and $\operatorname{SL}(n)$ invariant valuations on the space of $n$-dimensional convex bodies is established. As a consequence, complete characterizations of centro-affine and $L_p$ affine surface areas are obtained. The proofs make use of a new $\operatorname{SL}(n)$ shaping process for convex bodies.

Mathematical Subject Classification

Primary 2000: 52A20 Secondary: 52B45, 53A15

Milestones

Received: 16 January 2007
Revised: 9 May 2008
Accepted: 19 February 2009
Published online: 17 August 2010

Authors

Monika Ludwig

Institut für Diskrete Mathematik and Geometrie, TU Wien, Wiedner Hauptstrasse 8-10/104, 1040 Wien, Austria and Department of Mathematics, Polytechnic Institute of New York University, Six Metrotech Center, Brooklyn, New York 11201, United States

Matthias Reitzner

Fachbereich 6: Mathematik/Informatik, Universität Osnarbück, Albrechtstrasse 28a, 49076 Osnabrück, Germany