Abstract
In this paper, the classical dual mixed volume of star bodies V˜(K1,⋯,Kn) and dual Aleksandrov–Fenchel inequality are extended to the Orlicz space. Under the framework of dual Orlicz-Brunn-Minkowski theory, we put forward a new affine geometric quantity by calculating first order Orlicz variation of the dual mixed volume, and call it Orlicz multiple dual mixed volume. We generalize the fundamental notions and conclusions of the dual mixed volume and dual Aleksandrov-Fenchel inequality to an Orlicz setting. The classical dual Aleksandrov-Fenchel inequality and dual Orlicz-Minkowski inequality are all special cases of the new dual Orlicz-Aleksandrov-Fenchel inequality. The related concepts of Lp-dual multiple mixed volumes and Lp-dual Aleksandrov-Fenchel inequality are first derived here. As an application, the dual Orlicz–Brunn–Minkowski inequality for the Orlicz harmonic addition is also established.
Highlights
It is well known that vector addition is one of the important operators in convex geometry
Under the framework of dual Orlicz–Brunn–Minkowski theory, we introduce the affine geometric quantity by calculating the first order Orlicz variation of the dual mixed volumes, and call it Orlicz eφ (K1, . . . , Kn, Ln ), which involves (n + 1) star bodies in multiple dual mixed volumes, denoted by V
We find that Orlicz multiple dual mixed volume V
Summary
It is well known that vector addition is one of the important operators in convex geometry. Let S n denote the set of star bodies about the origin in Rn. The radial addition and volume are the core and essence of the classical dual. All the corresponding concepts and inequalities of the L p -space of the dual mixed volume V are all derived, which subverts the order of historical research on the issue, directly deriving the results of Orlicz space, saving a lot of time and resources. We extract the required geometric quantity, denoted by V and call it Orlicz multiple dual mixed volume of (n + 1) star bodies L1 , K1 , =, defined by eφ ( L1 , K1 , =) := φr0 (1) · d dε ε =0+. If φ is strictly convex, equality holds if and only if K and L are dilates
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