On maximal perimeters of convex sets with respect to measures

High Dimensional Seminar
Wednesday, September 4, 2019 - 3:00pm
1 hour (actually 50 minutes)
Skiles 006
Georgia Tech
We discuss the asymptotic value of the maximal perimeter of a convex set in an n-dimensional space with respect to certain classes of measures. Firstly, we derive a lower bound for this quantity for an arbitrary probability measure with first two moments bounded; the lower bound depends on the moments only. This lower bound is sharp in the case of the Gaussian measure (as was shown by Nazarov in 2001), and, more generally, in the case of rotation invariant log-concave measures (as was shown by the author in 2014). We show, that this lower bound is also sharp for a class of smooth log-concave measures satisfying certain uniform bounds on the hessian of the potential. In addition, we show a uniform upper bound of Cn for all isotropic log-concave measures, which is attained for the uniform distribution on the cube. Some improved bounds are also obtained for the Poisson density.