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

Aula Mancini, Scuola Normale Superiore

**Abstract.**

Given a Borel A in R^{n} 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 Freimanâ€™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.*