Affine function valued valuations

Abstract

A classification of SL$(n)$ contravariant, continuous function valued valuations on convex bodies is established. Such valuations are natural extensions of SL$(n)$ contravariant $L_p$ Minkowski valuations, the classification of which characterized $L_p$ projection bodies, which are fundamental in the $L_p$ Brunn-Minkowski theory, for $p\ge 1$. Hence our result will help to better understand extensions of the $L_p$ Brunn-Minkowski theory. In fact, our results characterize general projection functions which extend $L_p$ projection functions ($p$-th powers of the support functions of $L_p$ projection bodies) to projection functions in the $L_p$ Brunn-Minkowski theory for $0<p<1$ and in the Orlicz Brunn-Minkowski theory.

accepted by Int. Math. Res. Not

References in corpus (4)

• The Logarithmic Minkowski Problem
• Minkowski valuations on convex functions
• Laplace transforms and valuations
• $L_p$ Minkowski Valuations on polytopes

Cited by in corpus (1)

• $\mathrm{SL}(\mathrm{n})$ covariant function-valued valuations