Calculus of Variations and Geometric Measure Theory

A. Figalli

Quantitative stability results for the Brunn-Minkowski inequality

created by figalli on 25 Apr 2014
modified on 15 May 2014


Proceedings ICM 2014

Inserted: 25 apr 2014
Last Updated: 15 may 2014

Year: 2014


The Brunn-Minkowski inequality gives a lower bound of 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.