Fabian Mussnig

TU Wien | TU Wien · Institute of Discrete Mathematics and Geometry

A complete classification of all continuous, epi-translation and rotation invariant valuations on the space of super-coercive convex functions on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}...

Motivated by a problem for mixed Monge-Ampère measures of convex functions, we address a special case of a conjecture of Schneider and show that for every convex body $K$ the support of the mixed area measure $S(K[j], B_L^{n-1} [n-1-j], \cdot)$ is given by the set of $(K[ j], B_L^{n-1} [n-1-j])$-extreme unit normal vectors, where $B_L^{n-1}$ denote...

An introduction to geometric valuation theory is given. The focus is on classification results for SL(n) invariant and rigid motion invariant valuations on convex bodies and on convex functions.

An introduction to geometric valuation theory is given. The focus is on classification results for $\operatorname{SL}(n)$ invariant and rigid motion invariant valuations on convex bodies and on convex functions.

New proofs of the Hadwiger theorem for smooth and for continuous valuations on convex functions are obtained, and the Klain–Schneider theorem on convex functions is established. In addition, an extension theorem for valuations defined on functions with lower dimensional domain is proved, and its connection to the Abel transform is explained.

A complete family of functional Steiner formulas is established. As applications, an explicit representation of functional intrinsic volumes using special mixed Monge–Ampère measures and a new version of the Hadwiger theorem on convex functions are obtained.

New proofs of the Hadwiger theorem for smooth and for general valuations on convex functions are obtained, and the Klain-Schneider theorem on convex functions is established. In addition, an extension theorem for valuations defined on functions with lower dimensional domains is proved and its connection to the Abel transform is explained.

A complete family of functional Steiner formulas is established. As applications, an explicit representation of functional intrinsic volumes using special mixed Monge--Amp\`ere measures and a new version of the Hadwiger theorem on convex functions are obtained.

A new version of the Hadwiger theorem on convex functions is established and an explicit representation of functional intrinsic volumes is found using new functional Cauchy-Kubota formulas. In addition, connections between functional intrinsic volumes and their classical counterparts are obtained and non-negative valuations are classified.

We introduce a class of functional analogs of the symmetric difference metric on the space of coercive convex functions on ${\mathbb{R}}^n$ with full-dimensional domain. We show that convergence with respect to these metrics is equivalent to epi-convergence. For a large class of these natural metrics, we are able to provide a full classification of...

We introduce a class of functional analogs of the symmetric difference metric on the space of coercive convex functions on $\mathbb{R}^n$ with full-dimensional domain. We show that convergence with respect to these metrics is equivalent to epi-convergence. Furthermore, we give a full classification of all isometries with respect to some of the new...

A classification of SL(n) and translation covariant Minkowski valuations on log-concave functions is established. The moment vector and the recently introduced level set body of log-concave functions are characterized. Furthermore, analogs of the Euler characteristic and volume are characterized as SL(n) and translation invariant valuations on log-...

A complete classification of all continuous, epi-translation and rotation invariant valuations on the space of super-coercive convex functions on ${\mathbb R}^n$ is established. The valuations obtained are functional versions of the classical intrinsic volumes. For their definition, singular Hessian valuations are introduced.

The existence of a homogeneous decomposition for continuous and epi-translation invariant valuations on super-coercive functions is established. Continuous and epi-translation invariant valuations that are epi-homogeneous of degree n are classified. By duality, corresponding results are obtained for valuations on finite-valued convex functions.

\text{SL}(n)$ Invariant Valuations on Super-Coercive Convex Functions - Fabian Mussnig

The existence of a homogeneous decomposition for continuous and epi-translation invariant valuations on super-coercive functions is established. Continuous and epi-translation invariant valuations that are epi-homogeneous of degree $n$ are classified. By duality, corresponding results are obtained for valuations on finite-valued convex functions.

All non-negative, continuous, $\operatorname{SL}(n)$ and translation invariant valuations on the space of super-coercive, convex functions on $\mathbb{R}^n$ are classified. Furthermore, using the invariance of the function space under the Legendre transform, a classification of non-negative, continuous, $\operatorname{SL}(n)$ and dually translation...

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

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 ${\mathbb R}^n$.

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...

A classification of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {SL}}(n)$$\end{document}SL(n) contravariant Minkowski valuations on convex functions and a ch...

A classification of $\operatorname{SL}(n)$ and translation covariant Minkowski valuations on log-concave functions is established. The moment vector and the recently introduced level set body of log-concave functions are characterized. Furthermore, analogs of the Euler characteristic and volume are characterized as $\operatorname{SL}(n)$ and transl...

All continuous, SL$(n)$ and translation invariant valuations on the space of convex functions on ${\mathbb R}^n$ are completely classified.