Convexity of near-optimal orthogonal-pair-free sets on the unit sphere

A subset S of the unit sphere S² is called orthogonal-pair-free if and only if there do not exist two distinct points u, v ∈ S at distance π/2 from each other. Witsenhausen {witsenhausen} asked the following question: { What is the least upper bound α₃ on the Lesbegue measure of any measurable orthogonal-pair-free subset of S²?} We prove the following result in this paper: Let 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, α₃ = S ∈ A μ(S). Thus, if the double cap conjecture {kalai1} is not true, there is a set in A with measure strictly greater than the measure of the double cap.