In this paper, the well-known log-Minkowski inequality is extended to the Orlicz space. We first propose and establish an Orlicz logarithmic Minkowski inequality by introducing two new concepts of mixed volume measure and Orlicz mixed volume measure, and using the Orlicz Minkowski inequality for the mixed volumes. The Orlicz logarithmic Minkowski inequality in special case yields the Stancu's logarithmic Minkowski inequality. The $L_p$-mixed volume measure and $L_p$-logarithmic Minkowski inequality is first derived here, too.

中文翻译：

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Orlicz log-Minkowski不等式

在本文中，著名的log​​-Minkowski不等式被扩展到Orlicz空间。我们首先通过引入混合体积测度和Orlicz混合体积测度这两个新概念，并针对混合体积使用Orlicz Minkowski不等式，提出并建立Orlicz对数Minkowski不等式。在特殊情况下，Orlicz对数Minkowski不等式产生了Stancu的对数Minkowski不等式。的${\mathrm{大号}}_p$混合体积测量和 ${\mathrm{大号}}_p$-对数Minkowski不等式也首先在这里得到。