<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> be a bounded, convex set. A set <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {O} \subset \mathbb {R}^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> is an opaque set (for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>) if every line that intersects <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> also intersects <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {O}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">O</mi> </math></EquationSource> </InlineEquation>. What is the minimal possible length <i>L</i> of an opaque set? The best lower bound <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L \ge |\partial \Omega |/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo>≥</mo> <mo stretchy="false">|</mo> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">|</mo> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> is due to Jones (1962). It has been remarkably difficult to improve this bound, even in special cases where it is presumably very far from optimal. We prove a stability version: if <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L - |\partial \Omega |/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo>-</mo> <mo stretchy="false">|</mo> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">|</mo> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> is small, then any corresponding opaque set <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {O}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">O</mi> </math></EquationSource> </InlineEquation> has to be made up of curves whose tangents behave very much like the tangents of the boundary <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation> in a precise sense.</p>

A Stability Version of the Jones Opaque Set Inequality

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发表日期：

2026年5月6日

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Stefan Steinerberger

摘要

Let $\Omega \subset \mathbb {R}^2$ $Ω \subset R^{2}$ be a bounded, convex set. A set $\mathcal {O} \subset \mathbb {R}^2$ $O \subset R^{2}$ is an opaque set (for $\Omega $ $Ω$ ) if every line that intersects $\Omega $ $Ω$ also intersects $\mathcal {O}$ $O$ . What is the minimal possible length L of an opaque set? The best lower bound $L \ge |\partial \Omega |/2$ $L \geq | \partial Ω | / 2$ 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 $L - |\partial \Omega |/2$ $L - | \partial Ω | / 2$ is small, then any corresponding opaque set $\mathcal {O}$ $O$ has to be made up of curves whose tangents behave very much like the tangents of the boundary $\partial \Omega $ $\partial Ω$ in a precise sense.