Importance: Medium ✭✭
 Author(s): Kalai, Gil
 Subject: Combinatorics
 Keywords: combinatorial geometry independent set orthogonality projective plane sphere
 Posted by: Jon Noel on: September 15th, 2015
Conjecture   The largest measure of a Lebesgue measurable subset of the unit sphere of containing no pair of orthogonal vectors is attained by two open caps of geodesic radius 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 times the surface measure of the sphere. In , DeCorte and Pikhurko [DP] improved the multiplicative constant to . The conjecture above would imply that the measure is at most .

## Bibliography

[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? https://gilkalai.wordpress.com/2009/05/22/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.