Valuations on Log-Concave Functions

TL;DR

This paper classifies and characterizes valuations on log-concave functions that are covariant under special linear transformations and translations, extending geometric valuation concepts to a functional setting.

Contribution

It provides a comprehensive classification of SL(n) and translation covariant Minkowski valuations on log-concave functions, including moment vectors and level set bodies.

Findings

Characterization of the moment vector for log-concave functions

Identification of the level set body as a valuation

Analogues of Euler characteristic and volume for log-concave functions

Abstract

A classification of $\operatorname{SL}(n)$ and translation covariant Minkowski valuations on log-concave functions is established. The moment vector and the recently introduced level set body of log-concave functions are characterized. Furthermore, analogs of the Euler characteristic and volume are characterized as $\operatorname{SL}(n)$ and translation invariant valuations on log-concave functions.