Inserted: 7 feb 2013
Last Updated: 8 may 2014
Journal: J. Eur. Math. Soc. (JEMS)
Given a measurable set A\subset Rn 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.