Joseph Lehec (Université de Poitiers) – The thin-shell conjecture

Notes: online.

In a recent paper written jointly with Boaz Klartag, we prove that the variance of the Euclidean norm of any isotropic log-concave random vector is bounded above by a universal constant, not depending on the dimension. Thus, most of the mass of the random vector is concentrated in a thin spherical shell, whose width is order 1, while its radius is order root of the dimension. This confirms the thin-shell conjecture in high dimensional convex geometry. Our method relies on the construction of a certain coupling between log-affine perturbations of a given log-concave measure related to Eldan's stochastic localization and to the theory of non-linear filtering. Another ingredient is a recent breakthrough technique by Guan that was previously used in our proof of Bourgain's slicing conjecture, which is known to be implied by the thin-shell conjecture. In this talk, I'll first review the context and the history of the problem, before laying out the main steps of our proof.