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SUMMARY:Stability results for the semisum of sets in R^n - Alessio Figalli
DTSTART:20140519T140000Z
DTEND:20140519T150000Z
UID:TALK52734@talks.cam.ac.uk
CONTACT:37507
DESCRIPTION: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 s
 tatement is stable: if the measure of S is close to the one of A\, is A cl
 ose to his convex hull? More generally\, one may consider the semisum of t
 wo different sets A and B\, in which case our question corresponds to prov
 ing a stability result for the Brunn-Minkowski inequality. When n=1\, one 
 can approximate a set with finite unions of intervals to translate the pro
 blem to the integers Z. In this  discrete setting the question becomes a w
 ell-studied problem in additive combinatorics\, usually known as Freiman's
  Theorem. In this talk I will review some results in the one-dimensional d
 iscrete setting and describe how to answer to the problem in arbitrary dim
 ension.
LOCATION:CMS\, MR13
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