Marius Tiba

Sharp stability for the Brunn-Minkowski inequality for arbitrary sets

The Brunn-Minkowski inequality is a fundamental geometric inequality, closely related to the isoperimetric inequality. It states that for (open) sets A and B in R^d, we have |A+B|^{1/d} \geq |A|^{1/d}+|B|^{1/d}. Here A+B={a+b: a \in A, b \in B}. Equality holds if and only if A and B are convex and homothetic sets (one is a dilation of the other) in R^d. The stability of the Brunn-Minkowski inequality is the principle that if we are close to equality, then A and B must be close to being convex and homothetic. Using a combinatorial approach, we prove 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. This is joint work with Alessio Figalli and Peter van Hintum.