Valuations on Log-Concave Functions

Abstract

A classification of $\mathrm{SL}(n)$ 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 $\mathrm{SL}(n)$ and translation invariant valuations on log-concave functions.

References in corpus (7)

• Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems
• Valuations on Convex Functions
• Minkowski valuations on convex functions
• Laplace transforms and valuations
• Volume, Polar Volume and Euler Characteristic for Convex Functions
• $L_p$ Minkowski Valuations on polytopes
• Hessian valuations

Cited by in corpus (2)

• Valuations on Convex Bodies and Functions
• A Klain-Schneider Theorem for Vector-Valued Valuations on Convex Functions