A Stability Version of the Jones Opaque Set Inequality

TL;DR

This paper addresses the opaque-set problem for convex planar domains, focusing on the Jones inequality $L \geq \frac{|\,\partial\Omega\,|}{2}$ and its (long-standing) difficulty of improvement. It introduces an angular-measure framework by assigning to any opaque set $\mathcal{O}$ a measure $\mu_{\mathcal{O}}$ on $[0,2\pi)$ and to the boundary $\mu_{\partial\Omega}$, with total masses $L$ and $|\partial\Omega|/2$, respectively. The authors prove a quantitative stability bound in the Sobolev space $\dot H^{-2}(\mathbb{T})$: $\|\mu_{\mathcal{O}} - \mu_{\partial\Omega}\|_{\dot H^{-2}} \leq \frac{L^{1/4}}{\sqrt{2}}\left(L - \frac{|\partial\Omega|}{2}\right)^{3/4}$, linking near-optimal length to a near-equivalence of angular distributions via Fourier analysis. The results yield structural information about near-optimal opaque sets (e.g., for $\Omega=[0,1]^2$, the orientation distribution must be nearly axis-aligned) and suggest a broader geometric-analytic program to understand opaque sets through $\mathbb{R}^2 \times \mathbb{S}^1$ viewpoints and energy methods, with potential implications for disks and other shapes.

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.

Figures (4)

• Figure 1: Conjectured shortest opaque set for $[0,1]^2$ with length $\sqrt{2} + \sqrt{3/2} \sim 2.63$ (left) and opaque set for the unit disk (right).
• Figure 2: Conjectured shortest opaque set for the equilateral triangle with length $\sqrt{3}$ (left) and three sides of a rectangle (right).
• Figure 3: A set of line segments $\mathcal{O}$ (colored by angle) and the corresponding 'angular orientation' measure $\mu_{\mathcal{O}}$.
• Figure 4: Four corners with the top left being 'swept out'.

Theorems & Definitions (13)

• Theorem : Jones, 1962
• Theorem
• Proposition
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4