Hessian Valuations

Different approaches to introduce intrinsic volumes and more generally mixed volumes for convex and log-concave functions were proposed by Bobkov, Colesanti and Fragal\` a, by Rotem and Milman and by Alesker. They all turn out to be valuations on the corresponding spaces. Here 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))\,[{{D}^2} u(x)]_i\,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. Ultimately, a complete classification of continuous and rigid motion invariant valuations on this space of functions is the aim of this approach. The connection to Hadwiger's theorem will be discussed. The results presented in this talk are joint with Andrea Colesanti (University of Florence) and Fabian Mussnig (Technische Universit\"at Wien).

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