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, ${\alpha }_3={\mathrm{lim\hspace{0.17em}sup}}_{S\in \mathcal{A}}\mu (S)$. 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.