Minkowski valuations

Abstract

Centroid and difference bodies define $\operatorname {SL}(n)$ equivariant operators on convex bodies and these operators are valuations with respect to Minkowski addition. We derive a classification of $\operatorname {SL}(n)$ equivariant Minkowski valuations and give a characterization of these operators. We also derive a classification of $\operatorname {SL}(n)$ contravariant Minkowski valuations and of $L_p$-Minkowski valuations.