On the dual Orlicz mixed volumes

Jin, Hailin; Yuan, Shufeng; Leng, Gangsong

doi:10.1007/s11401-015-0920-x

Cited by 12 publications

(3 citation statements)

References 15 publications

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“…In particular, it is crucial for developing the theory of the dual Orlicz affine and geominimal surface areas [49] and the Orlicz-Legendre ellipsoids [65]. See e.g., [23,53,59] for more works in the rapidly developing dual Orlicz-Brunn-Minkowski theory.…”

Section: Introductionmentioning

confidence: 99%

The Dual Orlicz–Minkowski Problem

Zhu

Xing

2018

J Geom Anal

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In this paper, the dual Orlicz curvature measure is proposed and its basic properties are provided. A variational formula for the dual Orlicz-quermassintegral is established in order to give a geometric interpretation of the dual Orlicz curvature measure. Based on the established variational formula, a solution to the dual Orlicz-Minkowski problem regarding the dual Orlicz curvature measure is provided.2010 Mathematics Subject Classification: 53A15, 52B45, 52A39.

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Section: Introductionmentioning

confidence: 99%

The Dual Orlicz–Minkowski Problem

Zhu

Xing

2018

J Geom Anal

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“…In these papers the notions of -addition, -mixed volume, -affine surface area, -centroid body, andprojection body and inequalities were extended to an Orlicz setting. Advances in the theory and its dual can be found in [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48].…”

Section: Introductionmentioning

confidence: 99%

Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes

Zhao

2018

Journal of Function Spaces

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Our main aim is to generalize the classical mixed volumeV(K1,…,Kn)and Aleksandrov-Fenchel inequality to the Orlicz space. In the framework of Orlicz-Brunn-Minkowski theory, we introduce a new affine geometric quantity by calculating the Orlicz first-order variation of the mixed volume and call itOrlicz multiple mixed volumeof convex bodiesK1,…,Kn, andLn, denoted byVφ(K1,…,Kn,Ln), which involves(n+1)convex bodies inRn. The fundamental notions and conclusions of the mixed volume and Aleksandrov-Fenchel inequality are extended to an Orlicz setting. The related concepts and inequalities ofLp-multiple mixed volumeVp(K1,…,Kn,Ln)are also derived. The Orlicz-Aleksandrov-Fenchel inequality in special cases yieldsLp-Aleksandrov-Fenchel inequality, Orlicz-Minkowski inequality, and Orlicz isoperimetric type inequalities. As application, a new Orlicz-Brunn-Minkowski inequality for Orlicz harmonic addition is established, which implies Orlicz-Brunn-Minkowski inequalities for the volumes and quermassintegrals.

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“…A little weaker but useful definition for Orlicz addition was also provided by Xi, Jin and Leng [58] and independently in [8]; see following definition (1.6). For the Dual Orlicz-Brunn-Minkowski Theory, see [9,21,59]. For the Orlicz Minkowski problem, see [16,20].…”

Section: Introductionsmentioning

confidence: 99%

Orlicz valuations

Jin¹,

Leng²

2017

Indiana Univ. Math. J.

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In this paper, Orlicz valuations compatible with SL(n) transforms are classified. Unlike their L p analogs, the identity operator and the reflection operator are the only SL(n) compatible Orlicz valuations (up to dilations). It turns out that the Orlicz projection body operator, the Orlicz centroid body operator and the Orlicz difference body operator are not Orlicz valuations. The property that the Orlicz difference body operator is not an Orlicz valuation plays an important role in characterizing the identity operator and the reflection operator.2010 Mathematics Subject Classification. 52A20, 52B45.

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On the dual Orlicz mixed volumes

Abstract: In this paper, the authors define a harmonic Orlicz combination and a dual Orlicz mixed volume of star bodies, and then establish the dual Orlicz-Minkowski mixedvolume inequality and the dual Orlicz-Brunn-Minkowksi inequality.

Cited by 12 publications

References 15 publications

The Dual Orlicz–Minkowski Problem

The Dual Orlicz–Minkowski Problem

Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes

Orlicz valuations

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