Sharp stability of the Brunn-Minkowski inequality via optimal mass transportation

The Brunn-Minkowski inequality, applicable to bounded measurable sets A and B in Rᵈ, states that |A+B|1/d ≥ |A|1/d+|B|1/d. Equality is achieved if and only if A and B are convex and homothetic sets in Rᵈ. The concept of stability in this context concerns how, when approaching equality, sets A and B are close to homothetic convex sets. In a recent breakthrough [FvHT23], the authors of this paper proved the following folklore conjectures on the sharp stability for the Brunn-Minkowski inequality: (1) A linear stability result concerning the distance from A and B to their respective convex hulls. (2) A quadratic stability result concerning the distance from A and B to their common convex hull. As announced in [FvHT23], in the present paper, we leverage (1) in conjunction with a novel optimal transportation approach to offer an alternative proof for (2).