The Double Cap Conjecture

Importance: Medium ✭✭
Author(s): Kalai, Gil
Subject: Combinatorics
Recomm. for undergrads: no
Posted by: Jon Noel
on: September 15th, 2015

\begin{conjecture} The largest measure of a Lebesgue measurable subset of the unit sphere of $\mathbb{R}^n$ containing no pair of orthogonal vectors is attained by two open caps of geodesic radius $\pi/4$ around the north and south poles. \end{conjecture}

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The problem of determining the maximum was first considered by Witsenhausen [Wit] who proved that the measure of such a set is at most $\frac{1}{n}$ times the surface measure of the sphere. In $\mathbb{R}^3$, DeCorte and Pikhurko [DP] improved the multiplicative constant to $0.313< 1/3$. The conjecture above would imply that the measure is at most $1-1/\sqrt{2} \approx 0.2928$.


% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

[DP] E. DeCorte and O. Pikhurko, Spherical sets avoiding a prescribed set of angles, arXiv:1502.05030v2.

[Kalai] G. Kalai, How Large can a Spherical Set Without Two Orthogonal Vectors Be?

[Wit] H. S. Witsenhausen. Spherical sets without orthogonal point pairs. American Mathematical Monthly, pages 1101–1102, 1974.

* indicates original appearance(s) of problem.