$L_p$ Minkowski Valuations on polytopes

TL;DR

This paper provides a comprehensive classification of $L_p$ Minkowski valuations on polytopes for all $p \, \geq 1$, revealing new valuations in specific cases like $n=3$ and $p=1$.

Contribution

It extends previous classifications by removing additional conditions, covering all $p \geq 1$, including $p=\infty$, and identifying new valuations in certain dimensions and parameters.

Findings

Complete classification of $L_p$ Minkowski valuations on polytopes for $p \geq 1$.

Discovery of previously unknown valuations for $n=3$ and $p=1$.

Inclusion of the case $p=\infty$ in the classification.

Abstract

For $1 \leq p < \infty$, Ludwig, Haberl and Parapatits classified $L_p$ Minkowski valuations intertwining the special linear group with additional conditions such as homogeneity and continuity. In this paper,a complete classification of $L_p$ Minkowski valuations intertwining the special linear group on polytopes without any additional conditions is established for $p \geq 1$ including $p = \infty$. For $n=3$ and $p=1$, there exist valuations not mentioned before.