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Richard J. Gardner - One of the best experts on this subject based on the ideXlab platform.
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Gaussian Brunn-Minkowski inequalities
2020Co-Authors: Richard J. Gardner, Artem ZvavitchAbstract:Abstract. A detailed investigation is undertaken into Brunn-Minkowski-type inequalities for Gauss measure. A Gaussian dual Brunn-Minkowski Inequality is proved, together with precise equality conditions, and shown to be best possible from several points of view. A possible new Gaussian Brunn-Minkowski Inequality is proposed, and proved to be true in some special cases. Throughout the study attention is paid to precise equality conditions and conditions on the coefficients of dilatation. Interesting links are found to the S-Inequality and the (B) conjecture. An example is given to show that convexity is needed in the (B) conjecture
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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.
Gangsong Leng - One of the best experts on this subject based on the ideXlab platform.
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dar s conjecture and the log brunn Minkowski Inequality
Journal of Differential Geometry, 2016Co-Authors: Gangsong LengAbstract:In 1999, Dar conjectured that there is a stronger version of the celebrated Brunn-Minkowski Inequality. However, as pointed out by Campi, Gardner, and Gronchi in 2011, this problem seems to be open even for planar $o$-symmetric convex bodies. In this paper, we give a positive answer to Dar’s conjecture for all planar convex bodies. We also give the equality condition of this stronger Inequality. For planar $o$-symmetric convex bodies, the log–Brunn–Minkowski Inequality was established by Boroczky, Lutwak, Yang, and Zhang in 2012. It is stronger than the classical Brunn–Minkowski Inequality, for planar $o$-symmetric convex bodies. Gaoyong Zhang asked if there is a general version of this Inequality. Fortunately, the solution of Dar’s conjecture, especially, the definition of “dilation position”, inspires us to obtain a general version of the log–Brunn–Minkowski Inequality. As expected, this Inequality implies the classical Brunn–Minkowski Inequality for all planar convex bodies.
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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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brunn Minkowski Inequality for mixed intersection bodies
Journal of Mathematical Analysis and Applications, 2005Co-Authors: Chang-jian Zhao, Gangsong LengAbstract:Abstract Dual of the Brunn–Minkowski Inequality for mixed projection bodies are established for mixed intersection bodies.
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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.
Mutao Wang - One of the best experts on this subject based on the ideXlab platform.
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a Minkowski Inequality for hypersurfaces in the anti de sitter schwarzschild manifold
Communications on Pure and Applied Mathematics, 2016Co-Authors: Simon Brendle, Peiken Hung, Mutao WangAbstract:We prove a sharp Inequality for hypersurfaces in the n-dimensional anti-de Sitter-Schwarzschild manifold for general n>=3. This Inequality generalizes the classical Minkowski Inequality for surfaces in the three-dimensional euclidean space and has a natural interpretation in terms of the Penrose Inequality for collapsing null shells of dust. The proof relies on a new monotonicity formula for inverse mean curvature flow and uses a geometric Inequality established by the first author in [3].(c) 2015 Wiley Periodicals, Inc.
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a Minkowski Inequality for hypersurfaces in the anti de sitter schwarzschild manifold
Communications on Pure and Applied Mathematics, 2016Co-Authors: Simon Brendle, Peiken Hung, Mutao WangAbstract:We prove a sharp Inequality for hypersurfaces in the ndimensional Anti-deSitter-Schwarzschild manifold for general n ≥ 3. This Inequality generalizes the classical Minkowski Inequality for surfaces in the three dimensional Euclidean space, and has a natural interpretation in terms of the Penrose Inequality for collapsing null shells of dust. The proof relies on a new monotonicity formula for inverse mean curvature flow, and uses a geometric Inequality established by the first author in [3].
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a Minkowski Inequality for hypersurfaces in the anti desitter schwarzschild manifold
arXiv: Differential Geometry, 2012Co-Authors: Simon Brendle, Peiken Hung, Mutao WangAbstract:We prove a sharp Inequality for hypersurfaces in the n-dimensional Anti-deSitter-Schwarzschild manifold for general n greater or equal to 3. This Inequality generalizes the classical Minkowski Inequality for surfaces in the three dimensional Euclidean space, and has a natural interpretation in terms of the Penrose Inequality for collapsing null shells of dust. The proof relies on a new monotonicity formula for inverse mean curvature flow, and uses a geometric Inequality established by the first author in [3].
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.
Simon Brendle - One of the best experts on this subject based on the ideXlab platform.
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a Minkowski Inequality for hypersurfaces in the anti de sitter schwarzschild manifold
Communications on Pure and Applied Mathematics, 2016Co-Authors: Simon Brendle, Peiken Hung, Mutao WangAbstract:We prove a sharp Inequality for hypersurfaces in the n-dimensional anti-de Sitter-Schwarzschild manifold for general n>=3. This Inequality generalizes the classical Minkowski Inequality for surfaces in the three-dimensional euclidean space and has a natural interpretation in terms of the Penrose Inequality for collapsing null shells of dust. The proof relies on a new monotonicity formula for inverse mean curvature flow and uses a geometric Inequality established by the first author in [3].(c) 2015 Wiley Periodicals, Inc.
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a Minkowski Inequality for hypersurfaces in the anti de sitter schwarzschild manifold
Communications on Pure and Applied Mathematics, 2016Co-Authors: Simon Brendle, Peiken Hung, Mutao WangAbstract:We prove a sharp Inequality for hypersurfaces in the ndimensional Anti-deSitter-Schwarzschild manifold for general n ≥ 3. This Inequality generalizes the classical Minkowski Inequality for surfaces in the three dimensional Euclidean space, and has a natural interpretation in terms of the Penrose Inequality for collapsing null shells of dust. The proof relies on a new monotonicity formula for inverse mean curvature flow, and uses a geometric Inequality established by the first author in [3].
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a Minkowski Inequality for hypersurfaces in the anti desitter schwarzschild manifold
arXiv: Differential Geometry, 2012Co-Authors: Simon Brendle, Peiken Hung, Mutao WangAbstract:We prove a sharp Inequality for hypersurfaces in the n-dimensional Anti-deSitter-Schwarzschild manifold for general n greater or equal to 3. This Inequality generalizes the classical Minkowski Inequality for surfaces in the three dimensional Euclidean space, and has a natural interpretation in terms of the Penrose Inequality for collapsing null shells of dust. The proof relies on a new monotonicity formula for inverse mean curvature flow, and uses a geometric Inequality established by the first author in [3].