We shall discuss the state of the art on the problem of identifying the volume maximizing and minimizing hyperplane sections of p-balls. Specifically, we will present a recent work with P. Nayar (Warsaw) and T. Tkocz (CMU) identifying the volume maximizing section for p greater than a universal constant.

A century after the Khinchine inequalities were proven, most of the optimal constants in them are still unknown, but there is a slow, gradual progress in understanding their asymptotic behaviour. I will report on some recent result in this direction.

After reviewing the log-Brunn-Minkowski inequality and its infinitesimal form, we will present some considerations about a possible functional form of this inequality. This will lead us to a functional inequality related to the standard Poincaré inequality for the Gaussian measure.

We consider a very high dimensional torus $\T^n$ and a rather sparse lattice of points on it. Then if an analytic polynomial of degree $d<

The talk will be mostly based on a joint work with Bo'az Klartag from last year in which we prove a polylog estimate for the Kannan, Lovasz, Simonovits (KLS) conjecture. If time permits I will also discuss more recent improvements of the bound.