Quantitative stability for sumsets in $\mathbb R^n$ | Zendy

Alessio Figalli | Zendy; David Jerison | Zendy

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Quantitative stability for sumsets in $\mathbb R^n$

Author(s) -

Alessio Figalli,

David Jerison

Publication year - 2015

Publication title -

journal of the european mathematical society

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 3.549

H-Index - 64

eISSN - 1435-9863

pISSN - 1435-9855

DOI - 10.4171/jems/527

Subject(s) - mathematics , stability (learning theory) , combinatorics , statistics , computer science , machine learning

Given a measurable set $A\subset \R^n$ of positive measure, it is not difficult to show that $|A+A|=|2A|$ if and only if $A$ is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If $(|A+A|-|2A|)/|A|$ is small, is $A$ close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between $A$ and its convex hull in terms of $(|A+A|-|2A|)/|A|$.

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