*Accepted Paper*

**Inserted:** 7 feb 2013

**Last Updated:** 8 may 2014

**Journal:** J. Eur. Math. Soc. (JEMS)

**Year:** 2014

**Abstract:**

Given a measurable set A\subset R^{n} of positive measure, it is not difficult to show that the measure of A+A is equal to the measure of 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. 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 the difference between the measure of A+A and the one of 2A.

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