Convexity of near-optimal orthogonal-pair-free sets on the unit sphere
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909152561856512 |
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| author | Mudgal, Apurva |
| author_facet | Mudgal, Apurva |
| contents | A subset $S$ of the unit sphere $\mathbb{S}^2$ is called orthogonal-pair-free if and only if there do not exist two distinct points $u, v \in S$ at distance $\fracπ{2}$ from each other. Witsenhausen \cite{witsenhausen} asked the following question: {\it What is the least upper bound $α_3$ on the Lesbegue measure of any measurable orthogonal-pair-free subset of $\mathbb{S}^2$?} We prove the following result in this paper: Let $\mathcal{A}$ be the collection of all orthogonal-pair-free sets $S$ such that $S$ consists of a finite number of mutually disjoint convex sets. Then, $α_3 = \limsup_{S \in \mathcal{A}} μ(S)$. Thus, if the double cap conjecture \cite{kalai1} is not true, there is a set in $\mathcal{A}$ with measure strictly greater than the measure of the double cap. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18404 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convexity of near-optimal orthogonal-pair-free sets on the unit sphere Mudgal, Apurva Computational Geometry A subset $S$ of the unit sphere $\mathbb{S}^2$ is called orthogonal-pair-free if and only if there do not exist two distinct points $u, v \in S$ at distance $\fracπ{2}$ from each other. Witsenhausen \cite{witsenhausen} asked the following question: {\it What is the least upper bound $α_3$ on the Lesbegue measure of any measurable orthogonal-pair-free subset of $\mathbb{S}^2$?} We prove the following result in this paper: Let $\mathcal{A}$ be the collection of all orthogonal-pair-free sets $S$ such that $S$ consists of a finite number of mutually disjoint convex sets. Then, $α_3 = \limsup_{S \in \mathcal{A}} μ(S)$. Thus, if the double cap conjecture \cite{kalai1} is not true, there is a set in $\mathcal{A}$ with measure strictly greater than the measure of the double cap. |
| title | Convexity of near-optimal orthogonal-pair-free sets on the unit sphere |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2403.18404 |