Hessian valuations

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} ζ(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 $[D^{2}u{]}_i$ is the $i$th elementary symmetric function of the eigenvalues of the Hessian matrix, $D^{2}u$, of $u$. Under suitable assumptions on $\zeta$, these valuations are shown to be invariant under translations and rotations on convex and coercive functions.

30 pages