Calculus of Variations and Geometric Measure Theory

Stability results for the semisum of sets in R^n

Alessio Figalli (ETH Zurich)

created by ambrosio on 09 Dec 2013

17 dec 2013 -- 16:00   [open in google calendar]

Aula Mancini, Scuola Normale Superiore


Given a Borel A in Rn of positive measure, one can consider its semisum S=(A+A)2. It is clear that S contains A, and it is not diffi cult to prove that they have the same measure if and only if A is equal to his convex hull minus a set of measure zero. We now wonder whether this statement is stable: if the measure of S is close to the one of A, is A close to his convex hull? More generally, one may consider the semisum of two different sets A and B, in which case our question corresponds to proving a stability result for the Brunn-Minkowski inequality. When n=1, one can approximate a set with fi nite unions of intervals to translate the problem to the integers Z. In this discrete setting the question becomes a well-studied problem in additive combinatorics, usually known as Freiman’s Theorem. In this talk, which is intended for a general audience, I will review some results in the onedimensional discrete setting and show how to answer to the problem in arbitrary dimension.