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Orlicz log-Minkowski inequality. (English) Zbl 1462.52007

Summary: 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.

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A39 Mixed volumes and related topics in convex geometry
52A40 Inequalities and extremum problems involving convexity in convex geometry
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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