Victor-Emmanuel Brunel
CREST, Paris, France

Barycenters are a natural extension of the notion of linear averaging in metric spaces. We will present concentration inequalities for barycenters of i.i.d. random variables taking values in a metric spaces, under a condition of non-positive curvature. Examples of non-positively curved spaces are Euclidean and Hilbert spaces, metric trees, the cone of positive definite matrices equipped with a specific Riemannian metric (for which barycenters correspond to geometric means), etc. In particular, we will give extensions of Hoeffding’s and Bernstein’s inequalities. The talk will start with a short introduction on metric spaces, geodesics, curvature (in Alexandrov’s sense) and geodesic convexity, and will not require any prior knowledge on metric geometry.