Calculus of Variations and Geometric Measure Theory

A. Figalli - D. Jerison

Quantitative stability for the Brunn-Minkowski inequality

created by figalli on 22 Mar 2017
modified on 26 Apr 2017


Accepted Paper

Inserted: 22 mar 2017
Last Updated: 26 apr 2017

Journal: Adv. Math.
Year: 2017


The Brunn-Minkowski inequality gives a lower bound on the Lebesgue measure of a sumset in terms of the measures of the individual sets. This inequality plays a crucial role in the theory of convex bodies and has many interactions with isoperimetry and functional analysis. Stability of optimizers of this inequality in one dimension is a consequence of classical results in additive combinatorics. In this note we describe how optimal transportation and analytic tools can be used to obtain quantitative stability results in higher dimension.