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The Double Cap Conjecture
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.
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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 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.