Deep points in curved spaces

In this talk, based on a joint work with Shin-ichi Ohta (Osaka University) and Jordan Serres (Université Sorbonne, Paris), I will present a generalization of Grunbaum’s inequality to curved spaces. This inequality states that in Euclidean spaces, uniform distributions on convex bodies and, more generally, log-concave distributions, have deep points in Tukey’s sense: There always exists a point such that any half-space containing that point has mass bounded from below by some universal positive constant, independently of the dimension of the space. This fact is important in high dimensional statistics because it allows to discriminate deep points from shallow points even in high dimensions. It also has applications in convex optimization, e.g., to prove the algorithmic performance of cutting plane methods in high dimensions. 
We extend this inequality to non-Euclidean spaces that satisfy a curvature-dimension condition, where splitting theorems allow to extend the definition of half-spaces and reduce the analysis to one-dimensional computations.