Volume, Polar Volume and Euler Characteristic for Convex Functions

TL;DR

This paper characterizes functional analogs of classical geometric valuations such as volume, polar volume, and Euler characteristic for convex functions, establishing their invariance and continuity properties.

Contribution

It introduces a new analog of polar volume and characterizes these valuations as non-negative, continuous, SL(n) and translation invariant on convex functions.

Findings

Characterization of functional Euler characteristic and volume as valuations.

Introduction of a new polar volume analog for convex functions.

Proof of invariance and continuity properties of these valuations.

Abstract

Functional analogs of the Euler characteristic and volume together with a new analog of the polar volume are characterized as non-negative, continuous, $\operatorname{SL}(n)$ and translation invariant valuations on the space of finite, convex and coercive functions on ${\mathbb R}^n$.