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.

arXiv comment30 pages

References in corpus (4)

• Valuations on Convex Functions
• Minkowski valuations on convex functions
• Valuations on convex sets, non-commutative determinants, and pluripotential theory
• Monotone valuations on the space of convex functions

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• The Hadwiger theorem on convex functions, I
• Valuations on Log-Concave Functions
• Equivariant Endomorphisms of Convex Functions
• Valuations on Convex Bodies and Functions
• Additive kinematic formulas for convex functions
• Inequalities and counterexamples for functional intrinsic volumes and beyond
• A Klain-Schneider Theorem for Vector-Valued Valuations on Convex Functions
• The Legendre transform, the Laplace transform and valuations