  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.
