Dual curvature measures for log-concave functions

TL;DR

This paper introduces dual curvature measures for log-concave functions, extending concepts from convex geometry, and explores their properties, variational formulas, and a Minkowski problem in the symmetric case.

Contribution

It defines dual curvature measures for log-concave functions and establishes foundational results including variational formulas and conditions for a Minkowski problem.

Findings

Dual curvature measures generalize existing measures for convex bodies.

Variational formulas for these measures are derived.

Sufficient conditions for the Minkowski problem in the symmetric case are provided.

Abstract

We introduce dual curvature measures for log-concave functions, which in the case of characteristic functions recover the dual curvature measures for convex bodies introduced by Huang-Lutwak-Yang-Zhang in 2016. Variational formulas are shown. The associated Minkowski problem for these dual curvature measures is considered and sufficient conditions in the symmetric setting are demonstrated.