Affine function valued valuations

TL;DR

This paper classifies a broad class of valuations on convex bodies that extend key concepts in the $L_p$ and Orlicz Brunn-Minkowski theories, deepening understanding of these geometric frameworks.

Contribution

It provides a comprehensive classification of SL(n) contravariant, continuous function-valued valuations, extending the $L_p$ and Orlicz Brunn-Minkowski theories.

Findings

Classifies SL(n) contravariant, continuous function-valued valuations.

Characterizes extensions of $L_p$ projection functions for $0<p<1$ and Orlicz theory.

Enhances understanding of geometric valuations in convex geometry.

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 $p \geq 1$. 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.