The Brunn-Minkowski inequality states that for bounded measurable sets A and B in \mathbb{R}^n, we have |A+B|^{1/n} \geq |A|^{1/n}+|B|^{1/n}. Also, equality holds if and only if A and B are convex and homothetic sets in \mathbb{R}^d. The stability of this statement is a well-known problem that has attracted much attention in recent years. This paper gives a conclusive answer by proving the sharp stability result for the Brunn-Minkowski inequality on arbitrary sets.
