Four lectures by Apostolos Giannopoulos (National Technical University of Athens) and Joseph Lehec (Université de Poitiers).

• 10h-11h : A. Giannopoulos : The isotropic constant in the theory of high-dimensional convex bodies
• 11h-11h30 : coffee break
• 11h30-12h30 : A. Giannopoulos : The isotropic constant in the theory of high-dimensional convex bodies 2
• 12h30 - 14h : lunch
• 14h-14h50 : J. Lehec : solution to slicing conjecture 1
• 14h50-15h10 : coffee break
• 15h10-16h : J. Lehec : solution to the slicing conjecture 2

The ‘slicing’ or ‘hyperplane’ conjecture, put forward by J. Bourgain in the mid 80’s, is a fundamental question in geometric functional analysis that encodes deep interactions between measure and convexity in high dimensions. In its simplest form, it asserts that for any convex body of volume 1 in R^n, one can find a hyperplane section whose volume remains above an universal constant (independent of the dimension). An equivalent formulation amounts to bound the so-called ‘isoptropic constant’ associated to a convex body or to a log-concave measure.

This problem has structured many progresses in the geometry of convex bodies and log-concave measures in high dimension, along with concentration of measure and developments of probabilistic tools related to Gaussian processes. After several improvements in the recent years, the conjecture was recently solved by Joseph Lehec and Bo’az Klartag building on a crucial contribution by Qingyang Guan.

This day devoted to the slicing conjecture is intended to present the subject, the tools and its aftermath to non-experts. Apostolos Giannopoulos will introduce us to the problem, present the above-mentioned developments, and the various consequences of the conjecture. Joseph Lehec will present us its proof, and describe in particular the so-called stochastic localization method.

https://webusers.imj-prg.fr/~dario.cordero/ens/slicing