Quantitative stability for sumsets in Rⁿ

Given a measurable set A⊂ 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| .