Computer Science/Discrete Mathematics Seminar II

The Brunn-Minkowski inequality is a fundamental result in convex geometry controlling the volume of  the sum of subsets of $\mathbb{R}^n$. It asserts that for  sets $A,B\subset \mathbb{R}^n$ of equal volume and a parameter $t\in(0,1)$, we have $|tA+(1-t)B|\geq |A|$ with equality iff $A=B$ is convex.  Early work of Ruzsa, as well as results on the special case $A=B$ suggested a linear stability result; if $|tA+(1-t)B|\leq (1+\delta)|A|$, then $|co(A)\setminus A|=O_{n,t}(\delta)|A|$. On the other hand, the study of the stability of the isoperimetric inequality supported a folklore quadratic stability conjecture; for $\delta$ as before, we have (up to translation) $|A\triangle B|=O_{n,t}(\sqrt{\delta})|A|$. 

In this talk we explore the background of these conjectures, their connection with the discrete context of additive combinatorics (in particular the connection with geometric instances of the Polynomial Freiman-Ruzsa conjecture), and the proof of the linear conjecture. 

Joint work with Alessio Figalli and Marius Tiba.