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Károly J. Böröczky

Publications and source records attributed to Károly J. Böröczky.

At least 19 recordsLinked to original sources
arXiv2026

Centro-sectional measures for log-concave functions

We introduce centro-sectional measures with parameters q,m for log-concave functions on Rn, defined in terms of the q-th moments of their Radon transforms with respect to the Haar measure on m-dimensional subspaces, where m=1,...,n-1, and establish the corresponding variational formulas. Our measures generalize the notion of dual curvature measure if q=1, and are related to the Sine transform if q=2. In line with the coarea formula for log-concave functions as BV functions, the variational formulas give rise to the Euclidean centro-sectional measures and the spherical centro-sectional measures. In the symmetric setting, we solve the associated even functional centro-sectional Minkowski problem, which asks which pairs of measures can arise as the centro-sectional measures of an even log-concave function.

math.AP↗
arXiv2026

$L^2$-stability for the variance Brascamp-Lieb inequality

We prove an $L^2$-stability estimate for the variance Brascamp-Lieb inequality [J. Funct. Anal. 22 (4), 366-389 (1976)] by bootstrapping the recent $L^1$-stability theorem of Machado and Ramos [arXiv:2511.22636] under an additional assumption, which we call the super-Brascamp-Lieb inequality, of independent interest.

math.FA↗
arXiv2026

On Ball's conjectured Santaló type inequality

We prove that if $K$ is a symmetric and isotropic convex body in $\mathbb{R}^n$, then $$\int_K\langle x,u\rangle^2\,dx\int_{K^\circ}\langle x,u\rangle^2\,dx\leq \left(\int_{B_2^n}\langle x,u\rangle^2\,dx\right)^2,\qquad\forall u\in\mathbb{R}^n,$$with equality for some $u\neq o$, if and only if $K$ is a Euclidean ball. This confirms a conjecture by Keith Ball (1986), stating that for any symmetric convex body $K$ in $\mathbb{R}^n$, it holds $$\int_K\int_{K^\circ}\langle x,y\rangle^2\,dx\,dy\leq \int_{B_2^n}\int_{B_2^n}\langle x,y\rangle^2\,dx\,dy,$$with equality if and only if $K$ is an ellipsoid. Fortunately, our method for proving Ball's conjectured inequality admits a quantitative stability refinement, which in turn yields an asymptotically optimal stability version of the Blaschke-Santaló inequality for origin symmetric convex bodies in terms of the symmetric difference metric. This resolves another well known open problem.

math.MG↗
arXiv2026

Quantitative stability of extremal quasi conformal mappings

We establish quantitative stability results for classical distortion minimization problems in the theory of quasiconformal mappings. We consider the mean distortion functional and prove sharp stability estimates for the minimization problems regarding the linear stretch and spiral stretch maps, which arise as extremals in the class of mappins with finite distortion under appropriate boundary conditions. More precisely, we show that if a mapping has mean distortion close to the minimal value in the appropriate function class, then it must be quantitatively close, in certain Lebesgue norms.

math.CV↗
arXiv2026

A strengthening of the Blaschke-Santaló inequality for $o$-symmetric planar convex bodies

We verify the inequality $$ \frac{|K|}{|E|}+\frac{|K^*|}{|E^*|}\leq 2 $$ for any $o$-symmetric convex body $K\subset\mathbb{R}^2$ where $E$ is either the John ellipse of maximal area contained in $K$ or the minimal area Löwner ellipse containing $K$. The analogous estimate may not hold if $K$ is a planar but the assumption of $o$-symmetry is dropped, or if $K$ is $o$-symmetric convex body in $\mathbb{R}^n$ for $n\geq 3$. Our new inequality strengthens the Blaschke-Santaló inequality for $o$-symmetric convex bodies $K\subset\mathbb{R}^2$ with an error term of optimal order.

math.MG↗
arXiv2025

Approximate isoperimetry for convex polytopes

For all $n,ϕ\in \mathbb{N}$ with $ϕ\geqslant n+1$, the smallest possible isoperimetric quotient of an $n$-dimensional convex polytope that has $ϕ$ facets is shown to be bounded from above and from below by positive universal constant multiples of $\max\big\{n/\sqrt{1+\log (ϕ/n)},\sqrt{n}\big\}$. For all $n\in \mathbb{N}$ and $2n\leqslant β\in 2\mathbb{N}$, it is shown that every $n$-dimensional origin-symmetric convex polytope that has $β$ vertices admits an affine image whose isoperimetric quotient is at most a universal constant multiple of $\min\big\{\sqrt{\log(β/n)},n\big\}$, which is sharp. The weak isomorphic reverse isoperimetry conjecture is proved for $n$-dimensional convex polytopes that have $O(n)$ facets by demonstrating that any such polytope $K$ has an image $K'$ under a volume preserving matrix and a convex body $L\subseteq K'$ such that the isoperimetric quotient of $L$ is at most a universal constant multiple of $\sqrt{n}$, and also $\sqrt[n]{\mathrm{vol}_n(L)/\mathrm{vol}_n(K)}$ is at least a positive universal constant.

math.MG↗
arXiv2025

Dual Curvature Density Equation with Group Symmetry

This paper studies the general Lp dual curvature density equation under a group symmetry assumption. This geometric partial differential equation arises from the general Lp dual Minkowski problem of prescribing the Lp dual curvature measure of convex bodies. It is a Monge-Ampere type equation on the unit sphere. If the density function of the dual curvature measure is invariant under a closed subgroup of the orthogonal group, the geometric partial differential equation is solved in this paper for certain range of negative p using a variational method. This work generalizes recent results on the Lp dual Minkowski problem of origin-symmetric convex bodies.

math.AP↗
arXiv2025

The Logarithmic Minkowski Problem

In analogy with the classical Minkowski problem, necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone-volume measure of the unit ball of a finite dimensional Banach space.

math.MG↗
arXiv2024

The Brascamp-Lieb inequality in Convex Geometry and in the Theory of Algorithms

The Brascamp-Lieb inequality in harmonic analysis was proved by Brascamp and Lieb in the rank one case in 1976, and by Lieb in 1990. It says that in a certain inequality, the optimal constant can be determined by checking the inequality for centered Gaussian distributions. It was Keith M Ball's pioneering work around 1990 that led to various applications of the inequality in Convex Geometry, and even in Discrete Geometry, like Brazitikos' quantitative fractional version of the Helly Theorem. On the other hand, determining the optimal constant and possible Gaussian extremizers for the Brascamp-Lieb inequality can be formulated as a problem in terms of positive definite matrices, and this problem has intimate links to the Theory of Algorithms.

math.MG↗
arXiv2024

Strengthened inequalities for the mean width and the $\ell$-norm of origin symmetric convex bodies

Barthe, Schechtman and Schmuckenschläger proved that the cube maximizes the mean width of symmetric convex bodies whose John ellipsoid (maximal volume ellipsoid contained in the body) is the Euclidean unit ball, and the regular crosspolytope minimizes the mean width of symmetric convex bodies whose Löwner ellipsoid is the Euclidean unit ball. Here we prove close to be optimal stronger stability versions of these results, together with their counterparts about the $\ell$-norm based on Gaussian integrals. We also consider related stability results for the mean width and the $\ell$-norm of the convex hull of the support of even isotropic measures on the unit sphere.

math.MG↗
arXiv2024

Uniqueness when the $L_p$ curvature is close to be a constant for $p\in[0,1)$

For fixed positive integer $n$, $p\in[0,1]$, $a\in(0,1)$, we prove that if a function $g:\mathbb{S}^{n-1}\to \mathbb{R}$ is sufficiently close to 1, in the $C^a$ sense, then there exists a unique convex body $K$ whose $L_p$ curvature function equals $g$. This was previously established for $n=3$, $p=0$ by Chen, Feng, Liu \cite{CFL22} and in the symmetric case by Chen, Huang, Li, Liu \cite{CHLL20}. Related, we show that if $p=0$ and $n=4$ or $n\leq 3$ and $p\in[0,1)$, and the $L_p$ curvature function $g$ of a (sufficiently regular, containing the origin) convex body $K$ satisfies $λ^{-1}\leq g\leq λ$, for some $λ>1$, then $\max_{x\in\mathbb{S}^{n-1}}h_K(x)\leq C(p,λ)$, for some constant $C(p,λ)>0$ that depends only on $p$ and $λ$. This also extends a result from Chen, Feng, Liu \cite{CFL22}. Along the way, we obtain a result, that might be of independent interest, concerning the question of when the support of the $L_p$ surface area measure is lower dimensional. Finally, we establish a strong non-uniqueness result for the $L_p$-Minkowksi problem, for $-n<p<0$.

math.FA↗
arXiv2024

A quantitative stability result for the sphere packing problem in dimensions 8 and 24

We prove explicit stability estimates for the sphere packing problem in dimensions 8 and 24, showing that, in the lattice case, if a lattice is $\sim \varepsilon$ close to satisfying the optimal density, then it is, in a suitable sense, $O(\varepsilon^{1/2})$ close to the $E_8$ and Leech lattices, respectively. In the periodic setting, we prove that, under the same assumptions, we may take a large 'frame' through which our packing locally looks like $E_8$ or $Λ_{24}.$ Our methods make explicit use of the magic functions constructed by M. Viazovska in dimension 8 and by H. Cohn, A. Kumar, S. Miller, the second author, and M. Viazovska in dimension 24, together with results of independent interest on the abstract stability of the lattices $E_8$ and $Λ_{24}.$

math.MG↗
arXiv2023

Stability of optimal spherical codes

For many extremal configurations of points on a sphere, the linear programming approach can be used to show their optimality. In this paper we establish the general framework for showing stability of such configurations and use this framework to prove the stability of the two spherical codes formed by minimal vectors of the lattice $E_8$ and of the Leech lattice.

cs.IT↗
arXiv2022

Log-Brunn-Minkowski inequality under symmetry

We prove the log-Brunn-Minkowski conjecture for convex bodies with symmetries to $n$ independent hyperplanes, and discuss the equality case and the uniqueness of the solution of the related case of the logarithmic Minkowski problem. We also clarify a small gap in the known argument classifying the equality case of the log-Brunn-Minkowski conjecture for unconditional convex bodies.

math.FA↗