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Orlicz dual affine quermassintegrals. (English) Zbl 1395.52010

Summary: In the paper, our main aim is to generalize the dual affine quermassintegrals to the Orlicz space. Under the framework of Orlicz dual Brunn-Minkowski theory, we introduce a new affine geometric quantity by calculating the first-order variation of the dual affine quermassintegrals, and call it the Orlicz dual affine quermassintegral. The fundamental notions and conclusions of the dual affine quermassintegrals and the Minkoswki and Brunn-Minkowski inequalities for them are extended to an Orlicz setting, and the related concepts and inequalities of Orlicz dual mixed volumes are also included in our conclusions. The new Orlicz-Minkowski and Orlicz-Brunn-Minkowski inequalities in a special case yield the Orlicz dual Minkowski inequality and Orlicz dual Brunn-Minkowski inequality, which also imply the \(L_p\)-dual Minkowski inequality and Brunn-Minkowski inequality for the dual affine quermassintegrals.

MSC:

52A30 Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.)
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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