Hessian valuations

TL;DR

This paper introduces a new class of continuous valuations on convex functions involving Hessian eigenvalues, demonstrating invariance under transformations, and expanding the mathematical understanding of convex function valuations.

Contribution

It defines a novel class of valuations based on Hessian eigenvalues for convex functions and proves their invariance properties under geometric transformations.

Findings

Valuations are invariant under translations and rotations.

The valuations are well-defined on convex and coercive functions.

The paper extends the theory of valuations to include Hessian-based measures.

Abstract

A new class of continuous valuations on the space of convex functions on $\mathbb{R}^n$ is introduced. On smooth convex functions, they are defined for $i=0,\dots,n$ by \begin{equation*} u\mapsto \int_{\mathbb{R}^n} \zeta(u(x),x,\nabla u(x))\,[\operatorname{D}^2 u(x)]_i\,{\rm d} x \end{equation*} where $\zeta\in C(\mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^n)$ and $[\operatorname{D}^2 u]_i$ is the $i$th elementary symmetric function of the eigenvalues of the Hessian matrix, $\operatorname{D}^2 u$, of $u$. Under suitable assumptions on $\zeta$, these valuations are shown to be invariant under translations and rotations on convex and coercive functions.