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arXiv · 1811.10181

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

Shibing Chen·Yong Huang·Qi-rui Li·Jiakun Liu
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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 $C^α$ density, and the second one is for the Logarithmic Minkowski problem when the density of measure is a small $C^α$ even perturbation of the uniform density.

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Shibing Chen, Yong Huang, Qi-rui Li, Jiakun Liu. 2018-11-26. $L_p$-Brunn-Minkowski inequality for $p\in (1-\frac{c}{n^{\frac{3}{2}}}, 1)$. https://arxiv.org/abs/1811.10181

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