Stability for the Brunn-Minkowski and Riesz Rearrangement Inequalities, with Applications to Gaussian Concentration and Finite Range Non-local Isoperimetry
Printed: Oct 2017
We provide a simple, general argument to obtain improvements
of concentration-type inequalities starting from improvements
of their corresponding isoperimetric-type inequalities. We apply
this argument to obtain robust improvements of the Brunn-Minkowski
inequality (for Minkowski sums between generic sets and convex
sets) and of the Gaussian concentration inequality. The former
inequality is then used to obtain a robust improvement of the
Riesz rearrangement inequality under certain natural conditions.
These conditions are compatible with the applications to a finite-range
nonlocal isoperimetric problem arising in statistical mechanics.
Brunn-Minkowski inequality, Riesz rearrangement, Gaussian Concentration, Gates-Penrose-Lebowitz energy
49N99 - None of the above, but in this section