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We provide a simple, general argument to obtain improvements ofconcentration-type inequalities starting from improvements of theircorresponding isoperimetric-type inequalities. We apply this argument to obtainrobust improvements of the Brunn-Minkowski inequality (for Minkowski sumsbetween generic sets and convex sets) and of the Gaussian concentrationinequality. The former inequality is then used to obtain a robust improvementof the Riesz rearrangement inequality under certain natural conditions. Theseconditions are compatible with the applications to a finite-range nonlocalisoperimetric problem arising in statistical mechanics.

Stability for the Brunn-Minkowski and Riesz rearrangement inequalities,  with applications to Gaussian concentration and finite range non-local  isoperimetry