Quantitative stability of the Brunn-Minkowski inequality for sets of equal volume

Abstract

The authors prove a quantitative stability result for the Brunn-Minkowski inequality on sets of equal volume: If |A| = |B| > 0 and |A + B|1/n = (2+δ)|A|1/n for some small δ, then, up to a translation, both A and B are close (in terms of δ) to a convex set K. Although this result was already proved by the authors in a previous paper, the present paper provides a more elementary proof that the authors believe has its own interest. Also, the result here provides a stronger estimate for the stability exponent than the previous result of the authors.