Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2501.01004

A Stability Version of the Jones Opaque Set Inequality

Stefan Steinerberger
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $Ω\subset \mathbb{R}^2$ be a bounded, convex set. A set $O \subset \mathbb{R}^2$ is an opaque set (for $Ω$) if every line that intersects $Ω$ also intersects $O$. What is the minimal possible length $L$ of an opaque set? The best lower bound $L \geq |\partial Ω|/2$ is due to Jones (1962). It has been remarkably difficult to improve this bound, even in special cases where it is presumably very far from optimal. We prove a stability version: if $L - |\partial Ω|/2$ is small, then any corresponding opaque set $O$ has to be made up of curves whose tangents behave very much like the tangents of the boundary $\partial Ω$ in a precise sense.

Explore related subjects

math.MG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Stefan Steinerberger. 2025-01-02. A Stability Version of the Jones Opaque Set Inequality. https://arxiv.org/abs/2501.01004

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↗