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


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.


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