The following counterintuitive fact was conjectured by the third author and proved by Green [4]. It says that every finite transitive subset of a high dimensional sphere is close to some hyperplane. Here a subset X of a sphere in Rd is transitive if for every x, x′ ∈ X, there is some g ∈ O(Rd) so that gX = X and gx = x′. We say that X has width at most 2r if it lies within distance r of some hyperplane. The finiteness assumption is important since otherwise the whole sphere is a counterexample.

Abstract In 1878, Jordan showed that a finite subgroup of GL(n, ℂ) must possess an abelian normal subgroup whose index is bounded by a function of n alone. We will give the optimal bound for all n;… Expand

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… Expand

In these notes, the techniques aimed for obtaining explicit probability bounds in non-asymptotic theory of random matrices are described and some recent results are surveyed.Expand