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14/05/2013, 10:30 — 11:30 — Room P3.10, Mathematics Building

Alessio Figalli, *University of Texas at Austin*

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Stability results for sumsets in \(\mathbb{R}^n\)

Given a Borel set \(A\) 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'll review some results in the
one-dimensional discrete setting, and show how to answer to this
problem in arbitrary dimension.