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 Rⁿ for general \(n≥2\). We prove that, for every integer \(n≥2\), there exists a computable constant Cₙ>0 such that, for every 0<t≤12, there is a computable threshold εn,t>0 with the following property. Every pair of bounded Lebesgue-measurable sets A,B⊂ Rⁿ with \(|A|=|B|>0\) and 0≤δ:=|tA+(1-t)B|_*/|A|-1≤εn,t satisfies {equation*} |co(A) A|≤ C_nt⁻¹δ|A|, |co(B) B|≤ C_nt⁻⁽ⁿ⁻¹⁾δ|A|. {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 \(n≥3\). The planar case was established in [50]. Our argument can also be applied to the 2-dimensional case in a unified way.