Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2405.05954

The Gaussian measure of a convex body controls its maximal covering radius

Maud Szusterman
View original source ↗Read source document ↗Export metadata ↓

Abstract

The well-studied vector balancing constant $β(U, V)$ of a pair of convex bodies $(U,V)$, is lower bounded by a lattice counterpart, $α(U,V)$. In [BS97], Banaszczyk and Szarek proved that $α(B_2^n, V)\leq c$ when $V$ has Gaussian measure at least $\frac{1}{2}$, and conjectured that, for centrally symmetric $V$, $β(B_2^n, V)$ is always bounded by a function of the Gaussian measure of $V$, independent of $n$. We resolve this conjecture in the affirmative. Moreover, we show that the analogous result holds for $α(B_2^n, V)$ even without the central symmetry assumption.

Explore related subjects

math.MGmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Maud Szusterman. 2024-05-09. The Gaussian measure of a convex body controls its maximal covering radius. https://arxiv.org/abs/2405.05954

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

$β$-Uniform Convexity and Divisible Domains

Divisible convex sets have long been important in the study of Hilbert geometries. When a divisible convex set is an ellipsoid, the Hilbert geometry it induces is the hyperbolic space. In general, strictly convex divisible domains exhibit negative curvature properties, but only the ellipsoid is a CAT(0) space. The notion of p-uniform convexity from the theory of Banach spaces has been proposed by Shin-Ichi Ohta as a generalization of the Alexandrov-Toponogov comparison theorems to Finsler manifolds. We prove that a natural Finsler metric on a strictly convex divisible domain is $β$-uniformly convex, where the constant $β$ is related to the regularity of the boundary. We use this to show, with AI assistance, that the Hilbert metric, under suitable local and scale-dependent assumptions, is $β$-uniformly convex on such domains.

math.MG↗
arXiv2026

A positive solution to the $L^p$ projection centroid conjecture

In a classical paper [21] in 2000, Lutwak-Yang-Zhang established the $L^p$ analog of the Petty projection inequality and the $L^p$ analog of the Busemann-Petty centroid inequality. In Section 7 of [21], Lutwak-Yang-Zhang proposed the important $L^p$ projection centroid conjecture. We give a positive solution to the $L^p$ projection centroid conjecture in this work.

math.MG↗
arXiv2026

Minimal central slices of the regular simplex

We prove that minimal-volume hyperplane sections of the regular simplex through its centroid are parallel to a facet. The proof combines variational methods with Fourier-analytic techniques and zero-diminishing arguments to show that every critical normal vector has at most three distinct non-zero coordinates. Analysis of the two- and three-value cases then yields the sharp lower bound.

math.MG↗