We say that a finite subset of the unit sphere in is transitive if there is a group of isometries which acts transitively on it. We show that the width of any transitive set is bounded above by a constant times .
This is a consequence of the following result: if is a finite group and a unitary representation, and if is a unit vector, then there is another unit vector such that
These results answer a question of Yufei Zhao. An immediate consequence of our result is that the diameter of any quotient of the unit sphere by a finite group of isometries is at least .
"On the width of transitive sets: Bounds on matrix coefficients of finite groups." Duke Math. J. 169 (3) 551 - 578, 15 February 2020. https://doi.org/10.1215/00127094-2019-0074