Data e ora inizio evento:
Data e ora fine evento:
Sede:
Dipartimento di Matematica Guido Castelnuovo, Sapienza Università di Roma
Aula:
Sala di Consiglio
Speaker:
Alessio Figalli (The University of Texas at Austin)
Given a Borel A in \mathbb{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 difficult 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 in general, 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 finite unions of intervals to translate the problem onto \mathbb{Z}, and in the discrete setting this question becomes a well studied problem in additive combinatorics, usually known as Freiman's Theorem. In this talk I will review some results in the one-dimensional discrete setting, and show how to answer to this problem in arbitrary dimension.