Stability of the Brunn-Minkowski sumset inequality.

The Brunn-Minkowski inequality |A+B|^{1/k} >= |A|^{1/k}+|B|^{1/k} gives an optimal lower bound for the size of A+B given the sizes and A and B, with equality when A,B are homothetic convex sets. This talk will survey recent results and open questions surrounding the inverse question: How close are A,B to homothetic convex sets if we are close to equality?