On the width of transitive sets: bounds on matrix coefficients of finite groups

We say that a finite subset of the unit sphere in ${\mathbf{R}}^d$ 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 $(\log d{)}^{-1\mathrm{/}2}$. This is a consequence of the following result: If $G$ is a finite group and $\rho :G\to U_d(\mathbf{C})$ a unitary representation, and if $v\in {\mathbf{C}}^d$ is a unit vector, there is another unit vector $w \in $\mathbf{C}$^d$ such that $$\underset{g\in G}{\sup }\mathrm{\mid }⟨\rho (g)v,w⟩\mathrm{\mid }\le (1+c\log d{)}^{-1\mathrm{/}2}\mathrm{.}$$ These results answer a question of Yufei Zhao. An immediate consequence of our result is that the diameter of any quotient $S({\mathbf{R}}^d)\mathrm{/}G$ of the unit sphere by a finite group $G$ of isometries is at least $\pi/2 - o_{d \rightarrow \infty}(1)$.