The Double Cap Conjecture

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

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 $.


[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.