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Fabian Mussnig

Publications and source records attributed to Fabian Mussnig.

18 recordsLinked to original sources
arXiv2026

Explicit solutions to Christoffel-Minkowski problems and Hessian equations under rotational symmetries

An explicit solution to the Christoffel-Minkowski problem for convex bodies of revolution is presented. The conditions on the prescribed measure involve only first moments over spherical caps, and the support function of the resulting convex body is given by an explicit representation formula in terms of the measure. More generally, existence problems for mixed area measures are addressed. The approach relies on constructing explicit convex solutions to mixed Monge-Ampère equations on $\mathbb{R}^n$ under the assumption of radial symmetry, with the conditions on the measure being expressed through its values on open balls. As a special case, the Dirichlet problem for $k$-Hessian equations on $\mathbb{R}^n$ is treated.

math.MG↗
arXiv2026

Generalized outer linearizations and extremal properties of rotational epi-symmetrizations

We develop a functional extension of an extremal principle by Schneider (Monatsh. Math., 1967) by introducing generalized outer linearizations of convex functions. Given a coercive convex function on $\mathbb{R}^n$, a generalized outer linearization is defined as a convex minorant represented by a general but function-dependent set of slopes, thereby extending classical outer representations of convex bodies by supporting halfspaces. This representation converts geometric outer approximations by supporting halfspaces into functional approximations by supporting affine functions, and replaces outer normal data by a dual sampling problem in the domain of the Legendre--Fenchel transform. On a standard class of coercive convex functions, we derive a general extremal principle, showing that the rotational epi-symmetrization maximizes best approximations under outer linearizations of any monotone, concave functional that is upper semicontinuous with respect to epi-convergence. A central feature of the analysis is that it is carried out in the natural class of coercive, but not necessarily super-coercive, convex functions. Working in this setting introduces intricate topological and variational difficulties, which are addressed using refined duality and epi-convergence arguments. As an application of our main results, we derive a functional version of Urysohn's inequality, as well as an analytic extension of a classical covering result of Firey and Groemer (J. London Math. Soc., 1964). Finally, we prove an extremal inequality related to the piecewise affine approximation of convex functions.

math.FA↗
arXiv2026

Kubota-type formulas and supports of mixed measures

Kubota's integral formula expresses the intrinsic volumes of a convex body as averages over its projections onto linear subspaces. In this work, we introduce a new class of Kubota-type formulas for mixed area measures adapted to rotations around a fixed axis, which encode a crucial disintegration property. Our construction is motivated by applications to valuations on convex functions. In the latter framework, we obtain corresponding statements for (conjugate) mixed Monge-Ampère measures. As a by-product, we characterize supports of mixed area and mixed Monge-Ampère measures, thereby confirming a special case of a conjecture by Schneider.

math.MG↗
arXiv2025

The Vectorial Hadwiger Theorem on Convex Functions

A complete classification of continuous, dually epi-translation invariant, and rotation equivariant valuations on convex functions is established. This characterizes the recently introduced functional Minkowski vectors, which naturally extend the classical Minkowski relations. For this, the existence of these operators with singular densities is shown, along with additional representations involving mixed Monge-Ampère measures, Kubota-type formulas, and area measures of higher dimensional convex bodies. Dual results are formulated for valuations on super-coercive convex functions.

math.MG↗
arXiv2025

A Klain-Schneider Theorem for Vector-Valued Valuations on Convex Functions

A functional analog of the Klain-Schneider theorem for vector-valued valuations on convex functions is established, providing a classification of continuous, translation covariant, simple valuations. Under additional rotation equivariance assumptions, an analytic counterpart of the moment vector is characterized alongside a new epi-translation invariant valuation. The former arises as the top-degree operator in a family of functional intrinsic moments, which are linked to functional intrinsic volumes through translations. The latter represents the top-degree operator in a class of Minkowski vectors, which are introduced in this article and which lack classical counterparts on convex bodies, as they vanish due to the Minkowski relations. Additional classification results are obtained for homogeneous valuations of extremal degrees.

math.MG↗
arXiv2024

Inequalities and counterexamples for functional intrinsic volumes and beyond

We show that analytic analogs of Brunn-Minkowski-type inequalities fail for functional intrinsic volumes on convex functions. This is demonstrated both through counterexamples and by connecting the problem to results of Colesanti, Hug, and Saorín Gómez. By restricting to a smaller set of admissible functions, we then introduce a family of variational functionals and establish Wulff-type inequalities for these quantities. In addition, we derive inequalities for the corresponding family of mixed functionals, thereby generalizing an earlier Alexandrov-Fenchel-type inequality by Klartag and recovering a special case of a recent Pólya-Szegő-type inequality by Bianchi, Cianchi, and Gronchi.

math.FA↗
arXiv2024

Additive kinematic formulas for convex functions

We prove a functional version of the additive kinematic formula as an application of the Hadwiger theorem on convex functions together with a Kubota-type formula for mixed Monge-Ampère measures. As an application, we give a new explanation for the equivalence of the representations of functional intrinsic volumes as singular Hessian valuations and as integrals with respect to mixed Monge-Ampère measures. In addition, we obtain a new integral geometric formula for mixed area measures of convex bodies, where integration on $\operatorname{SO}(n-1)\times \operatorname{O}(1)$ is considered.

math.MG↗
arXiv2023

The Hadwiger theorem on convex functions, II: Cauchy-Kubota formulas

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.

math.FA↗
arXiv2023

Valuations on Convex Bodies and 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.

math.MG↗
arXiv2022

The Hadwiger theorem on convex functions, I

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.

math.FA↗
arXiv2022

The Hadwiger theorem on convex functions, IV: The Klain approach

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 domains is proved, and its connection to the Abel transform is explained.

math.FA↗
arXiv2020

Metrics and Isometries for Convex Functions

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 metrics. Moreover, we introduce two new functional analogs of the Hausdorff metric on the spaces of coercive convex functions and super-coercive convex functions, respectively, and prove equivalence to epi-convergence.

math.FA↗
arXiv2019

$\operatorname{SL}(n)$ invariant valuations on super-coercive 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 invariant valuations is obtained. In both cases, different functional analogs of the Euler characteristic, volume and polar volume are characterized.

math.MG↗
arXiv2018

Volume, Polar Volume and Euler Characteristic for Convex Functions

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 ${\mathbb R}^n$.

math.MG↗
arXiv2017

Valuations on Log-Concave Functions

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 translation invariant valuations on log-concave functions.

math.MG↗
arXiv2017

Minkowski valuations on convex functions

A classification of SL$(n)$ contravariant Minkowski valuations on convex functions and a characterization of the projection body operator are established. The associated LYZ measure is characterized. In addition, a new SL$(n)$ covariant Minkowski valuation on convex functions is defined and characterized.

math.FA↗
arXiv2017

Valuations on Convex Functions

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

math.FA↗