Orlicz-Aleksandrov-Fenchel inequality for Orlicz multiple mixed volumes
type: Item
instance of: scholarly article
title: Orlicz-Aleksandrov-Fenchel inequality for Orlicz multiple mixed volumes
zbMATH Open document ID: 1402.52010
DOI: 10.1155/2018/9752178
author: Changjian Zhao
published in: Journal of Function Spaces
publication date: 2018-11-15T00:00:00Z
review text: Summary: Our main aim is to generalize the classical mixed volume \(V(K_1, \ldots, K_n)\) 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 it \textit{Orlicz multiple mixed volume} of convex bodies \(K_1, \ldots, K_n\), and \(L_n\), denoted by \(V_\varphi(K_1, \ldots, K_n, L_n)\), which involves \((n + 1)\) convex bodies in \(\mathbb{R}^n\). The fundamental notions and conclusions of the mixed volume and Aleksandrov-Fenchel inequality are extended to an Orlicz setting. The related concepts and inequalities of \(L_p\)-multiple mixed volume \(V_p(K_1, \ldots, K_n, L_n)\) are also derived. The Orlicz-Aleksandrov-Fenchel inequality in special cases yields \(L_p\)-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.
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A38
zbMATH DE Number: 6979885
zbMATH Keywords: Orlicz space
zbMATH Keywords: Orlicz multiple mixed volume
zbMATH Keywords: Orlicz-Brunn-Minkowski inequality
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2018/9752178
OpenAlex ID: W2891267137
cites work: Polar Means of Convex Bodies and a Dual to the Brunn-Minkowski Theorem
cites work: $p$-Means of Convex Bodies.
cites work: The \(L^p\)-Busemann-Petty centroid inequality
cites work: On the reverse L p –busemann–petty centroid inequality
cites work: Volume Inequalities forLp‐Zonotopes
cites work: The \(L_p\)-Minkowski problem and the Minkowski problem in centroaffine geometry
cites work: The Centro-Affine Hadwiger Theorem
cites work: Asymmetric affine \(L_p\) Sobolev inequalities
cites work: General \(L_{p}\) affine isoperimetric inequalities
cites work: An asymmetric affine Pólya-Szegő principle
cites work: \(\mathrm{SL}(n)\) invariant valuations on polytopes
cites work: The Brunn-Minkowski-Firey theory. II: Affine and geominimal surface areas
cites work: On the $L_{p}$-Minkowski problem
cites work: $L_p$ John Ellipsoids
cites work: \(L_ p\) affine isoperimetric inequalities.
cites work: Sharp affine \(L_ p\) Sobolev inequalities.
cites work: The Brunn-Minkowski-Firey inequality for nonconvex sets
cites work: SL(n)-covariantLp-Minkowski valuations
cites work: $\mathrm {SL}(n)$-contravariant $L_p$-Minkowski valuations
cites work: Surface bodies and \(p\)-affine surface area
cites work: Rényi divergence and \(L_p\)-affine surface area for convex bodies
cites work: New \(L_p\) affine isoperimetric inequalities
cites work: Orlicz projection bodies
cites work: Orlicz centroid bodies
cites work: The Orlicz-Brunn-Minkowski theory: a general framework, additions, and inequalities
cites work: The Orlicz Brunn-Minkowski inequality
cites work: The Orlicz centroid inequality for star bodies
cites work: The even Orlicz Minkowski problem
cites work: General affine surface areas
cites work: Inequalities for general mixed affine surface areas
cites work: Affine isoperimetric inequalities in the functional Orlicz-Brunn-Minkowski theory
cites work: Orlicz-Brunn-Minkowski inequalities for Blaschke-Minkowski homomorphisms
cites work: On the Orlicz Minkowski problem for polytopes
cites work: On the dual Orlicz mixed volumes
cites work: Q3272715
cites work: A new proof of the Orlicz Busemann-Petty centroid inequality
cites work: Affine Orlicz Pólya-Szegő principle for log-concave functions
cites work: Q4000065
cites work: The planar Orlicz Minkowski problem in the \(L^1\)-sense
cites work: Dual mixed Orlicz-Brunn-Minkowski inequality and dual Orlicz mixed quermassintegrals
cites work: Dar's conjecture and the log-Brunn-Minkowski inequality
cites work: Orlicz dual mixed volumes
cites work: Q4596969
cites work: Orlicz-Brunn-Minkowski inequality for radial Blaschke-Minkowski homomorphisms
cites work: Orlicz dual affine quermassintegrals
cites work: Q4645531
cites work: Orlicz mean dual affine quermassintegrals
cites work: Dual Orlicz-Brunn-Minkowski theory
cites work: The Brunn-Minkowski-Firey theory. I: Mixed volumes and the Minkowski problem
cites work: Convex Bodies The Brunn-MinkowskiTheory
cites work: Q5769211
cites work: Q4753788
cites work: Q4039749
cites work: Q5593766
cites work: Q3862690
cites work: Q3884743
cites work: Q3409343
cites work: Mixed Projection Inequalities
cites work: Volume of Mixed Bodies
cites work: Q4323630
Wikidata QID: Q129234069
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