arXiv subjects

Leo Brauner

Publications and source records attributed to Leo Brauner.

5 recordsLinked to original sources

arXiv2025

Mixed Christoffel-Minkowski problems for bodies of revolution

The mixed Christoffel-Minkowski problem asks for necessary and sufficient conditions for a Borel measure on the Euclidean unit sphere to be the mixed area measure of some convex bodies, one of which, appearing multiple times, is free and the rest are fixed. In the case where all bodies involved are symmetric around a common axis, we provide a complete solution to this problem, without assuming any regularity. In particular, we refine Firey's classification of area measures of figures of revolution. In our argument, we introduce an easy way to transform mixed area measures and mixed volumes involving axially symmetric bodies, and we significantly improve Firey's estimate on the local behavior of area measures. As a secondary result, we obtain a family of Hadwiger type theorems for convex valuations that are invariant under rotations around an axis.

math.MG↗

arXiv2025

The Christoffel problem for the disk area measure

The mixed Christoffel problem asks for necessary and sufficient conditions for a Borel measure on the Euclidean unit sphere to be the mixed area measure of some convex bodies, all but one of them are fixed. We consider the case in which the reference bodies are $(n-1)$-dimensional disks lying in a fixed hyperplane. We obtain an integral representation that reconstructs the support function of a convex body from its disk area measure, without any regularity assumptions. In the smooth setting, we reformulate the problem as a linear differential equation on the sphere, and derive a necessary and sufficient condition on the density of the disk area measure guaranteeing both convexity and regularity of the solution.

math.MG↗

arXiv2024

The Klain approach to zonal valuations

We show an analogue of the Klain-Schneider theorem for valuations that are invariant under rotations around a fixed axis, called zonal. Using this, we establish a new integral representation of zonal valuations involving mixed area measures with a disk. In our argument, we introduce an easy way to translate between this representation and the one involving area measures, yielding a shorter proof of a recent characterization by Knoerr. As applications, we obtain various zonal integral geometric formulas, extending results by Hug, Mussnig, and Ulivelli. Finally, we provide a simpler proof of the integral representation of the mean section operators by Goodey and Weil.

math.MG↗

arXiv2024

Lefschetz operators on convex valuations

We investigate the action of Alesker's Lefschetz operators on translation invariant valuations on convex bodies. For scalar valued valuations, we describe this action on the level of Klain-Schneider functions by a Radon type transform, generalizing a result by Schuster and Wannerer. In the case of rotationally equivariant Minkowski valuations, the Lefschetz operators act on the generating function as a convolution transform. We show that the convolution kernel satisfies a Legendre type differential equation, and thus, is a strictly positive function that is smooth up to one point.

math.MG↗

arXiv2023

Fixed Points of Mean Section Operators

We characterize rotation equivariant bounded linear operators from $C(\mathbb{S}^{n-1})$ to $C^2(\mathbb{S}^{n-1})$ by the mass distribution of the spherical Laplacian of their kernel function on small polar caps. Using this characterization, we show that every continuous, homogeneous, translation invariant, and rotation equivariant Minkowski valuation $\Phi$ that is weakly monotone maps the space of convex bodies with a $C^2$ support function into itself. As an application, we prove that if $\Phi$ is in addition even or a mean section operator, then Euclidean balls are its only fixed points in some $C^2$ neighborhood of the unit ball. Our approach unifies and extends previous results by Ivaki from 2017 and the second author together with Schuster from 2021.

math.MG↗