A Combinatorial Perspective on Geometric Inequalities

Seminar in Mathematics

The Brunn-Minkowski inequality is an important result in convex geometry and analysis, closely related to the isoperimetric inequality. It states that for (open) sets A and B in $\mathbb{R}^d$, we have $|A+B|^{1/d} \geq |A|^{1/d}+|B|^{1/d}$. Here $A+B=\{x+y : x \in A, y \in B\}$. Equality holds if and only if $A$ and $B$ are convex and homothetic sets (one is a dilation of the other) in $\mathbb{R}^d$. The stability of the Brunn-Minkowski inequality is the principle that if we are \emph{close} to equality, then A and B must be \emph{close} to being convex and homothetic. In this talk, we present a sharp stability result for the Brunn-Minkowski inequality, establishing the exact dependency between the two notions of closeness, thus concluding a long line of research on this problem. We shall also discuss sharp stability results for discrete analogues of the Brunn-Minkowski inequality. These are motivated by the fundamental \emph{inverse sumset problem} in additive combinatorics: if the size of $A+B$ is \emph{small}, what can we say about the structure of the sets $A$ and $B$? This talk is mostly based on joint work with Alessio Figalli and Peter van Hintum.