Convexity of near-optimal orthogonal-pair-free sets on the unit sphere
<|reference_start|>Convexity of near-optimal orthogonal-pair-free sets on the unit sphere: 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 $\frac{\pi}{2}$ from each other. Witsenhausen \cite{witsenhausen} asked the following question: {\it What is the least upper bound $\alpha_3$ 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, $\alpha_3 = \limsup_{S \in \mathcal{A}} \mu(S)$. 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.<|reference_end|>
@article{mudgal2024convexity,
    title={Convexity of near-optimal orthogonal-pair-free sets on the unit sphere},
    author={Apurva Mudgal},
    journal={arXiv preprint arXiv:2403.18404},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.18404},
    primaryClass={cs.CG}
}