By studying $L^p$-combinations of strongly isomorphic polytopes, we give a simple proof of the equivalence of the $L^p$-Brunn-Minkowski inequality conjectured by Böröczky, Lutwak, Yang and Zhang to the local version of the inequality studied by Colesanti, Livshyts, and Marsiglietti and by Kolesnikov and Milman. In addition, we prove the local inequality in dimension 2 for $p=0$, yielding a new proof of the log-Brunn-Minkowski inequality in the plane.

中文翻译：

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L ^p -Brunn-Minkowski不等式的局部和全局版本的等价性

通过学习 ${\mathrm{大号}}^p$-强同构多态的组合，我们给出了等价的简单证明 ${\mathrm{大号}}^p$-Böröczky，Lutwak，Yang和Zhang推测的-Brunn-Minkowski不等式与Colesanti，Livshyts和Marsiglietti以及Kolesnikov和Milman研究的当地不等式有关。此外，我们证明了2维的局部不等式$p=0$，这提供了平面上log-Brunn-Minkowski不等式的新证明。