Quantitative stability for sumsets in ${ℝ}^n$

Figalli, Alessio, and Jerison, David. "Quantitative stability for sumsets in $\mathbb {R}^n$." Journal of the European Mathematical Society 017.5 (2015): 1079-1106. <http://eudml.org/doc/277337>.

@article{Figalli2015,
abstract = {Given a measurable set $A\subset \mathbb \{R\}^n$ 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 $(|A+A|-|2A|)/|A|$.},
author = {Figalli, Alessio, Jerison, David},
journal = {Journal of the European Mathematical Society},
keywords = {quantitative stability; sumsets; Freiman’s theorem; sumsets; quantitative stability; Freiman's theorem},
language = {eng},
number = {5},
pages = {1079-1106},
publisher = {European Mathematical Society Publishing House},
title = {Quantitative stability for sumsets in $\mathbb \{R\}^n$},
url = {http://eudml.org/doc/277337},
volume = {017},
year = {2015},
}

TY - JOUR
AU - Figalli, Alessio
AU - Jerison, David
TI - Quantitative stability for sumsets in $\mathbb {R}^n$
JO - Journal of the European Mathematical Society
PY - 2015
PB - European Mathematical Society Publishing House
VL - 017
IS - 5
SP - 1079
EP - 1106
AB - Given a measurable set $A\subset \mathbb {R}^n$ 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 $(|A+A|-|2A|)/|A|$.
LA - eng
KW - quantitative stability; sumsets; Freiman’s theorem; sumsets; quantitative stability; Freiman's theorem
UR - http://eudml.org/doc/277337
ER -