A Stability Version of the Jones Opaque Set Inequality
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916547633610752 |
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| author | Steinerberger, Stefan |
| author_facet | Steinerberger, Stefan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01004 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Stability Version of the Jones Opaque Set Inequality Steinerberger, Stefan Metric Geometry 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. |
| title | A Stability Version of the Jones Opaque Set Inequality |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2501.01004 |