Dimensionality and the stability of the Brunn-Minkowski inequality
  We prove stability estimates for the Brunn-Minkowski inequality for convex sets. Unlike existing stability results, our estimates improve as the dimension grows. Our results are equivalent to a thin shell bound, which is one of the central ingredients in the proof of the central limit theorem for convex sets. 
Ronen Eldan and Bo`az Klartag
math.MG math.PR