A remark on A+B and A-A for compact sets in R^n
  We prove in particular that if A be a compact convex subset of R^n, and B from R^n be an arbitrary compact set then \mu (A-A) \ll \mu(A+B)^2 / (\sqrt{n} \mu (A)), provided that \mu(B)\ge \mu(A). 
Tomasz Schoen and Ilya D. Shkredov
math.CO math.NT