Orlicz valuations

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.