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CATEGORIES:Geometric Analysis and Partial Differential Equati
ons seminar
SUMMARY:Stability results for the semisum of sets in R^n -
Alessio Figalli
DTSTART;TZID=Europe/London:20140519T150000
DTEND;TZID=Europe/London:20140519T160000
UID:TALK52734AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/52734
DESCRIPTION:Given a Borel A in R^n of positive measure\, one c
an consider its semisum S=(A+A)/2. It is clear tha
t S contains A\, and it is not difficult to prove
that they have the same measure if and only if A i
s equal to his convex hull minus a set of measure
zero. We now wonder whether this statement is stab
le: if the measure of S is close to the one of A\,
is A close to his convex hull? More generally\, o
ne may consider the semisum of two different sets
A and B\, in which case our question corresponds t
o proving a stability result for the Brunn-Minkows
ki inequality. When n=1\, one can approximate a se
t with finite unions of intervals to translate the
problem to the integers Z. In this discrete sett
ing the question becomes a well-studied problem in
additive combinatorics\, usually known as Freiman
's Theorem. In this talk I will review some result
s in the one-dimensional discrete setting and desc
ribe how to answer to the problem in arbitrary dim
ension.
LOCATION:CMS\, MR13
CONTACT:
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