Orlicz valuations

TL;DR

This paper classifies Orlicz valuations compatible with SL(n) transformations, revealing that only the identity and reflection operators qualify, unlike their Lp counterparts, and excludes several known Orlicz operators from being valuations.

Contribution

It provides a complete classification of SL(n)-compatible Orlicz valuations, highlighting the uniqueness of the identity and reflection operators in this context.

Findings

Only the identity and reflection operators are SL(n)-compatible Orlicz valuations.

The Orlicz projection, centroid, and difference body operators are not Orlicz valuations.

The non-valuation property of the Orlicz difference body operator is key to the classification.

Abstract

In this paper, Orlicz valuations compatible with $SL(n)$ transforms are classified. Unlike their $L_p$ analogs, the identity operator and the reflection operator are the only $SL(n)$ compatible Orlicz valuations (up to dilations). It turns out that the Orlicz projection body operator, the Orlicz centroid body operator and the Orlicz difference body operator are not Orlicz valuations. The property that the Orlicz difference body operator is not an Orlicz valuation plays an important role in characterizing the identity operator and the reflection operator.