Sharp Quantitative Stability of the Brunn-Minkowski Inequality in Rⁿ with n ≥ 2

In this paper, we are concerned with the sharp quantitative stability with optimal exponents of the Brunn-Minkowski inequality in \(R^n\) for general \(n\geq2\). We prove that, for every integer \(n\ge2\), there exists a computable constant \(C_n>0\) such that, for every \(0<t\le\frac12\), there is a computable threshold \(\varepsilon_{n,t}>0\) with the following property. Every pair of bounded Lebesgue-measurable sets \(A,B\subset R^n\) with \(|A|=|B|>0\) and \(0\le\delta:=|tA+(1-t)B|_*/|A|-1\le\varepsilon_{n,t}\) satisfies \begin{equation*} |co(A)\setminus A|\le C_nt^{-1}\delta|A|, \qquad |co(B)\setminus B|\le C_nt^{-(n-1)}\delta|A|. \end{equation*} Both the powers -1 and \(-(n-1)\) of \(t\) in the above two estimates are optimal and can be attained. Our result prove the Conjecture 14.1 of Figalli, van Hintum and Tiba [25] for all the higher dimension cases \(n\geq3\). The planar case was established in [50]. Our argument can also be applied to the 2-dimensional case in a unified way.