Searcharxiv⌕ Search

arXiv subjects

Karoly J. Boroczky

Publications and source records attributed to Karoly J. Boroczky.

At least 19 recordsLinked to original sources
arXiv2026

The Weighted Dual Minkowski Problem Under Group Symmetry

The paper studies the weighted dual Minkowski problem for the weighted $q$th dual curvature measure of convex bodies in $\mathbb R^n$ first posed by Huang, Lutwak, Yang, and Zhang. Most of the results are in the G-invariant setting, i.e. when the convex bodies are invariant under a closed subgroup $G$ of $O(n)$. They generalize previous results on the existence of solutions to the dual Minkowski problem for origin-symmetric convex bodies. It is proved: (i) a full characterization of the G-invariant weighted dual curvature measure for q\in(0,1], (ii) a necessary and sufficient condition for the existence of solutions to the G-invariant weighted dual Minkowski problem for q\in(1,n), (iii) a sufficient condition for the G-invariant weighted dual curvature measure for q>1, and (iv) an extension of Henk-Pollehn's dual curvature measure concentration property.

math.AP↗
arXiv2026

The Lp centro-sectional Minkowski problem

As part of Lutwak's broadening of the Brunn-Minkowski theory, and extending the notion of affine quermassintegrals and dual curvature measure discussed by Milman, Yehudayoff and Huang, Lutwak, Yang and Zhang, centro-sectional measures with real parameter q have been recently introduced by Cai, Leng, Wu, Xi. In this paper, we introduce the Lp cross sectional Minkowski problem analogously to the Lp dual Minkowski problem formulated by Lutwak, Yang and Zhang. We solve the Lp centro-sectional Minkowski problem for p>1 and q>0, discuss the regularity and uniqueness of the solution, and prove Lp Brunn-Minkowski-type inequalities when $p$ is relatively large.

math.AP↗
arXiv2025

Exponential valuations on lattice polygons valued at formal power series

We classify valuations on lattice polygons with values in the ring of formal power series that commute with the action of the affine unimodular group. A typical example of such valuations is induced by the Laplace transform, but as it turns out there are many more. The classification is done in terms of formal power series that satisfy certain functional equations. We align our classification with the decomposition into so-called dilative components.

math.MG↗
arXiv2025

Pal's isominwidth problem in the hyperbolic space

The paper focuses on possible hyperbolic versions of the classical Pal isominwidth inequality in R^2 from 1921, which states that for a fixed minimal width, the regular triangle has minimal area. We note that the isominwidth problem is still wide open in R^n for n>2. Recent work on the isominwidth problem on the sphere S^2 shows that the solution is the regular spherical triangle when the width is at most π/2 according to Bezdek and Blekherman, while Freyer and Sagmeister proved that the minimizer is the polar of a spherical Reuleaux triangle when the minimal width is greater than π/2. In this paper, the hyperbolic isominwidth problem is discussed with respect to the probably most natural notion of width due to Lassak in the hyperbolic space H^n where strips bounded by a supporting hyperplane and a corresponding hypersphere are considered. On the one hand, we show that the volume of a convex body of given minimal Lassak width w>0 in H^n might be arbitrarily small; therefore, the isominwidth problem for convex bodies in H^n does not make sense. On the other hand, in the two-dimensional case, we prove that among horocyclically convex bodies of given Lassak width in H^2, the area is minimized by the regular horocyclic triangle.

math.MG↗
arXiv2025

Compactness of the $L_p$ dual Minkowski problem in $\mathbb{R}^3$

We prove the $C^0$ estimate for the $L_p$ $q$th dual Minkowski problem on $S^2$ under fairly general conditions; namely, when $p$ lies in [0,1) and $q>2+p$, and the $L_p$ $q$th dual curvarture is bounded and bounded away from zero. We note that it is known that the analogous $C^0$ estimate does not hold if $p<-1$ and $q=3$. As a corollary of our $C^0$ estimate, we deduce the uniqueness of the solution of the near isotropic $q$th $L_p$ dual Minkowski problem on $S^2$ if $q$ is close to 3 and the $q$th $L_p$ dual curvature is Holder close to be the constant one function.

math.AP↗
arXiv2025

Exponential valuations on lattice polygons

We classify translatively exponential and GL(2,Z) covariant valuations on lattice polygons valued at measurable real functions. A typical example of such valuations is induced by the Laplace transform, but as it turns out there are many more. The argument uses the ergodicity of the linear action of SL(2,Z) on R2, and some elementary properties of the Fibonacci numbers.

math.NT↗
arXiv2024

The Logarithmic Minkowski conjecture and the $L_p$-Minkowski Problem

The current state of art concerning the $L_p$ Minkowski problem as a Monge-Ampere equation on the sphere and Lutwak's Logarithmic Minkowski conjecture about the uniqueness of even solution in the $p=0$ case are surveyed and connections to many related problems are discussed.

math.AP↗
arXiv2022

Convex bodies of constant width in spaces of constant curvature and the extremal area of Reuleaux triangles

Extending Blaschke and Lebesgue's classical result in the Euclidean plane, it has been recently proved in spherical and the hyperbolic cases, as well, that Reuleaux triangles have the minimal area among convex domains of constant width $D$. We prove an essentially optimal stability version of this statement in each of the three types of surfaces of constant curvature. In addition, we summarize the fundamental properties of convex bodies of constant width in spaces of constant curvature, and provide a characterization in the hyperbolic case in terms of horospheres.

math.MG↗
arXiv2016

Strengthened volume inequalities for L_p zonoids of even isotropic measures

We strengthen the volume inequalities for L_p zonoids of even isotropic measures and for their duals, which are due to Ball, Barthe and Lutwak, Yang, Zhang. Along the way, we prove a stronger version of the Brascamp-Lieb inequality for a family of functions that can approximate arbitrary well some Gaussians when equality holds. The special case p=\infty yields a stability version of the reverse isoperimetric inequality for centrally symmetric bodies.

math.PR↗
arXiv2016

Minkowski valuations on lattice polytopes

A complete classification is established of Minkowski valuations on lattice polytopes that intertwine the special linear group over the integers and are translation invariant. In the contravariant case, the only such valuations are multiples of projection bodies. In the equivariant case, the only such valuations are generalized difference bodies combined with multiples of the newly defined discrete Steiner point.

math.MG↗
arXiv2015

The discrete logarithmic Minkowski problem

Existence of solution of the logarithmic Minkowski problem is proved for the case where the discrete measures on the unit sphere satisfy the subspace concentration condition with respect to some special proper subspaces. In order to understand how optimal this condition is, we discuss certain conditions that any cone volume measure satisfies.

math.MG↗