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 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. http://cvgmt.sns.it/seminar/332/
When
Tue Dec 17, 2013 3pm – 4pm Coordinated Universal Time
