ON THE WIDTH OF TRANSITIVE SETS: BOUNDS ON MATRIX COEFFICIENTS OF FINITE GROUPS

We say that a finite subset of the unit sphere in R-d is transitive if there is a group of isometrics which acts transitively on it. We show that the width of any transitive set is bounded above by a constant times (log d)(-1/2). This is a consequence of the following result: if G is a finite group and rho : G -> U-d(C) a unitary representation, and if v is an element of C(d )is a unit vector, then there is another unit vector w is an element of C-d such that Sup(g is an element of G)vertical bar vertical bar <= (1 + c log d)(-1/2). These results answer a question of Yufei Zhao. An immediate consequence of our result is that the diameter of any quotient S (R-d)/G of the unit sphere by a finite group G of isometrics is at least pi/2 - o(d) (-> infinity) (1).