$L_p$-Brunn-Minkowski inequality for $p\in (1-\frac{c}{n^{\frac{3}{2}}}, 1)$

Abstract: Kolesnikov-Milman [9] established a local $L_p$-Brunn-Minkowski inequality for $p\in(1-c/n^{\frac{3}{2}},1).$ Based on their local uniqueness results for the $L_p$-Minkowski problem, we prove in this paper the (global) $L_p$-Brunn-Minkowski inequality. Two uniqueness results are also obtained: the first one is for the $L_p$-Minkowski problem when $p\in (1-c/n^{\frac{3}{2}}, 1)$ for general measure with even positive $C^{\alpha}$ density, and the second one is for the Logarithmic Minkowski problem when the density of measure is a small $C^{\alpha}$ even perturbation of the uniform density.