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Richard J Gardner - One of the best experts on this subject based on the ideXlab platform.
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the dual orlicz brunn Minkowski theory
Journal of Mathematical Analysis and Applications, 2015Co-Authors: Richard J Gardner, Wolfgang Weil, Deping YeAbstract:This paper introduces the dual Orlicz-Brunn-Minkowski theory for star sets. A radial Orlicz addition of two or more star sets is proposed and a corresponding dual Orlicz- Brunn-Minkowski inequality is established. Based on a radial Orlicz linear combination of two star sets, a formula for the dual Orlicz mixed volume is derived and a corresponding dual Orlicz-Minkowski inequality proved. The inequalities proved yield as special cases the precise duals of the conjectured log-Brunn-Minkowski and log-Minkowski inequalities of Boroczky, Lutwak, Yang, and Zhang. A new addition of star sets called radial M-addition is also introduced and shown to relate to the radial Orlicz addition.
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the orlicz brunn Minkowski theory a general framework additions and inequalities
arXiv: Metric Geometry, 2013Co-Authors: Richard J Gardner, Wolfgang WeilAbstract:The Orlicz-Brunn-Minkowski theory, introduced by Lutwak, Yang, and Zhang, is a new extension of the classical Brunn-Minkowski theory. It represents a generalization of the $L_p$-Brunn-Minkowski theory, analogous to the way that Orlicz spaces generalize $L_p$ spaces. For appropriate convex functions $\varphi:[0,\infty)^m\to [0,\infty)$, a new way of combining arbitrary sets in $\R^n$ is introduced. This operation, called Orlicz addition and denoted by $+_{\varphi}$, has several desirable properties, but is not associative unless it reduces to $L_p$ addition. A general framework is introduced for the Orlicz-Brunn-Minkowski theory that includes both the new addition and previously introduced concepts, and makes clear for the first time the relation to Orlicz spaces and norms. It is also shown that Orlicz addition is intimately related to a natural and fundamental generalization of Minkowski addition called $M$-addition. The results obtained show, roughly speaking, that the Orlicz-Brunn-Minkowski theory is the most general possible based on an addition that retains all the basic geometrical properties enjoyed by the $L_p$-Brunn-Minkowski theory. Inequalities of the Brunn-Minkowski type are obtained, both for $M$-addition and Orlicz addition. The new Orlicz-Brunn-Minkowski inequality implies the $L_p$-Brunn-Minkowski inequality. New Orlicz-Minkowski inequalities are obtained that generalize the $L_p$-Minkowski inequality. One of these has connections with the conjectured log-Brunn-Minkowski inequality of Lutwak, Yang, and Zhang, and in fact these two inequalities together are shown to split the classical Brunn-Minkowski inequality.
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the brunn Minkowski inequality
Bulletin of the American Mathematical Society, 2002Co-Authors: Richard J GardnerAbstract:In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The Brunn-Minkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of Rn, and deserves to be better known. This guide explains the relationship between the Brunn-Minkowski inequality and other inequalities in geometry and analysis, and some applications.
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The Brunn–Minkowski Inequality, Minkowski's First Inequality, and Their Duals☆☆☆
Journal of Mathematical Analysis and Applications, 2000Co-Authors: Richard J Gardner, Salvatore Flavio VassalloAbstract:Abstract Quantitative versions are given of the equivalence of the Brunn–Minkowski inequality and Minkowski's first inequality from the Brunn–Minkowski theory. Similar quantitative versions are obtained of the equivalence of the corresponding inequalities from Lutwak's dual Brunn–Minkowski theory. The main results are shown to be the best possible up to constant factors.
Gangsong Leng - One of the best experts on this subject based on the ideXlab platform.
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On the discrete Orlicz Minkowski problem II
Geometriae Dedicata, 2019Co-Authors: Yuchi Wu, Dongmeng Xi, Gangsong LengAbstract:The Orlicz Minkowski problem is a generalization of the $$L_p$$ Minkowski problem. For a class of appropriate functions and discrete measures that have no essential subspaces, the existence is demonstrated for the discrete Orlicz Minkowski problem. This is a non-trivial extension of the discrete $$L_p$$ Minkowski problem for $$p
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Orlicz–Brunn–Minkowski inequalities for Blaschke–Minkowski homomorphisms
Geometriae Dedicata, 2016Co-Authors: Feixiang Chen, Gangsong LengAbstract:Orlicz–Brunn–Minkowski type inequalities for Blaschke–Minkowski homomorphisms and their polars are established.
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the orlicz brunn Minkowski inequality
Advances in Mathematics, 2014Co-Authors: Dongmeng Xi, Gangsong LengAbstract:Abstract The Orlicz Brunn–Minkowski theory originated with the work of Lutwak, Yang, and Zhang in 2010. In this paper, we first introduce the Orlicz addition of convex bodies containing the origin in their interiors, and then extend the L p Brunn–Minkowski inequality to the Orlicz Brunn–Minkowski inequality. Furthermore, we extend the L p Minkowski mixed volume inequality to the Orlicz mixed volume inequality by using the Orlicz Brunn–Minkowski inequality.
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The Orlicz Brunn–Minkowski inequality ☆
Advances in Mathematics, 2014Co-Authors: Dongmeng Xi, Gangsong LengAbstract:Abstract The Orlicz Brunn–Minkowski theory originated with the work of Lutwak, Yang, and Zhang in 2010. In this paper, we first introduce the Orlicz addition of convex bodies containing the origin in their interiors, and then extend the L p Brunn–Minkowski inequality to the Orlicz Brunn–Minkowski inequality. Furthermore, we extend the L p Minkowski mixed volume inequality to the Orlicz mixed volume inequality by using the Orlicz Brunn–Minkowski inequality.
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The Brunn-Minkowski inequality for volume differences
Advances in Applied Mathematics, 2004Co-Authors: Gangsong LengAbstract:In this paper, we establish some theorems for the volume differences of compact domains, which are extensions of the Brunn-Minkowski inequality, Minkowski inequality, and isoperimetric inequality. Further, we give a generalizations of the matrix form of the Brunn-Minkowski inequality and prove the Brunn-Minkowski inequality for quermassintegral differences of convex bodies.
Deping Ye - One of the best experts on this subject based on the ideXlab platform.
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the dual orlicz brunn Minkowski theory
Journal of Mathematical Analysis and Applications, 2015Co-Authors: Richard J Gardner, Wolfgang Weil, Deping YeAbstract:This paper introduces the dual Orlicz-Brunn-Minkowski theory for star sets. A radial Orlicz addition of two or more star sets is proposed and a corresponding dual Orlicz- Brunn-Minkowski inequality is established. Based on a radial Orlicz linear combination of two star sets, a formula for the dual Orlicz mixed volume is derived and a corresponding dual Orlicz-Minkowski inequality proved. The inequalities proved yield as special cases the precise duals of the conjectured log-Brunn-Minkowski and log-Minkowski inequalities of Boroczky, Lutwak, Yang, and Zhang. A new addition of star sets called radial M-addition is also introduced and shown to relate to the radial Orlicz addition.
Wolfgang Weil - One of the best experts on this subject based on the ideXlab platform.
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the dual orlicz brunn Minkowski theory
Journal of Mathematical Analysis and Applications, 2015Co-Authors: Richard J Gardner, Wolfgang Weil, Deping YeAbstract:This paper introduces the dual Orlicz-Brunn-Minkowski theory for star sets. A radial Orlicz addition of two or more star sets is proposed and a corresponding dual Orlicz- Brunn-Minkowski inequality is established. Based on a radial Orlicz linear combination of two star sets, a formula for the dual Orlicz mixed volume is derived and a corresponding dual Orlicz-Minkowski inequality proved. The inequalities proved yield as special cases the precise duals of the conjectured log-Brunn-Minkowski and log-Minkowski inequalities of Boroczky, Lutwak, Yang, and Zhang. A new addition of star sets called radial M-addition is also introduced and shown to relate to the radial Orlicz addition.
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the orlicz brunn Minkowski theory a general framework additions and inequalities
arXiv: Metric Geometry, 2013Co-Authors: Richard J Gardner, Wolfgang WeilAbstract:The Orlicz-Brunn-Minkowski theory, introduced by Lutwak, Yang, and Zhang, is a new extension of the classical Brunn-Minkowski theory. It represents a generalization of the $L_p$-Brunn-Minkowski theory, analogous to the way that Orlicz spaces generalize $L_p$ spaces. For appropriate convex functions $\varphi:[0,\infty)^m\to [0,\infty)$, a new way of combining arbitrary sets in $\R^n$ is introduced. This operation, called Orlicz addition and denoted by $+_{\varphi}$, has several desirable properties, but is not associative unless it reduces to $L_p$ addition. A general framework is introduced for the Orlicz-Brunn-Minkowski theory that includes both the new addition and previously introduced concepts, and makes clear for the first time the relation to Orlicz spaces and norms. It is also shown that Orlicz addition is intimately related to a natural and fundamental generalization of Minkowski addition called $M$-addition. The results obtained show, roughly speaking, that the Orlicz-Brunn-Minkowski theory is the most general possible based on an addition that retains all the basic geometrical properties enjoyed by the $L_p$-Brunn-Minkowski theory. Inequalities of the Brunn-Minkowski type are obtained, both for $M$-addition and Orlicz addition. The new Orlicz-Brunn-Minkowski inequality implies the $L_p$-Brunn-Minkowski inequality. New Orlicz-Minkowski inequalities are obtained that generalize the $L_p$-Minkowski inequality. One of these has connections with the conjectured log-Brunn-Minkowski inequality of Lutwak, Yang, and Zhang, and in fact these two inequalities together are shown to split the classical Brunn-Minkowski inequality.
Chang-jian Zhao - One of the best experts on this subject based on the ideXlab platform.
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Orlicz-Brunn-Minkowski inequality for radial Blaschke-Minkowski homomorphisms
Quaestiones Mathematicae, 2018Co-Authors: Chang-jian ZhaoAbstract:In this paper, we extend the Brunn-Minkowski inequality for radial Blaschke-Minkowski homomorphisms to an Orlicz setting and an Orlicz-Brunn- Minkowski inequality for radial Blaschke-Minkowski homomorphisms is established. The new Orlicz-Brun-Minkowski inequality in special case yields the Lp-Brunn- Minkowski inequality for the radial mixed Blaschke-Minkowski homomorphisms and the mixed intersection bodies, respectively. Mathematics Subject Classication (2010): 52A20, 52A40. Key words: Orlicz radial addition, intersection body, radial Blaschke-Minkowski homo- morphisms, Orlicz dual mixed volume, Orlicz dual Brunn-Minkowski inequality.
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On Blaschke–Minkowski Homomorphisms and Radial Blaschke–Minkowski Homomorphisms
Journal of Geometric Analysis, 2015Co-Authors: Chang-jian ZhaoAbstract:In this paper we establish Minkowski, Brunn–Minkowski, and Aleksandrov–Fenchel type inequalities for volume differences of Blaschke–Minkowski homomorphisms and radial Blaschke–Minkowski homomorphisms.
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On radial Blaschke-Minkowski homomorphisms
Geometriae Dedicata, 2012Co-Authors: Chang-jian ZhaoAbstract:In this article we first establish a dual isoperimetric type inequality for mixed radial Blaschke-Minkowski homomorphisms of different orders. Second, we prove a dual Brunn-Minkowski-type and a dual Minkowski-type inequality for mixed radial Blaschke-Minkowski homomorphisms.
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On Blaschke-Minkowski homomorphisms
Geometriae Dedicata, 2010Co-Authors: Chang-jian ZhaoAbstract:In this article we establish a Brunn-Minkowski-type inequality for mixed Blaschke-Minkowski homomorphisms.
