  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.
