In this paper, we define a dual \(\phi \)-combination \({\widetilde{Q}}_{\phi , \xi }\). Using the log-convexity of strictly decreasing convex function \(\phi ^{-1}\), we give the dual \(\phi \)-Brunn–Minkowski inequality. Moreover, the equivalence between the dual \(\phi \)-Brunn–Minkowski inequality and the dual \(\phi \)-Minkowski mixed volume inequality is demonstrated.

中文翻译：

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对偶 $$\phi $$ ϕ -Brunn–Minkowski 不等式

在本文中，我们定义了一个对偶\(\phi \) -组合\({\widetilde{Q}}_{\phi , \xi }\)。使用严格递减凸函数\(\phi ^{-1}\) 的对数凸性，我们给出了对偶\(\phi \) -Brunn-Minkowski 不等式。此外，证明了对偶\(\phi \) -Brunn-Minkowski 不等式和对偶\(\phi \) -Minkowski 混合体积不等式之间的等价性。