scholarly journals The dual Brunn–Minkowski inequality for log-volume of star bodies

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Dandan Lai ◽  
Hailin Jin
Keyword(s):  
Minkowski Inequality ◽  
Star Bodies

AbstractThis paper aims to consider the dual Brunn–Minkowski inequality for log-volume of star bodies, and the equivalent Minkowski inequality for mixed log-volume.

Orlicz Addition for Measures and an Optimization Problem for the -divergence

2019 ◽  
Vol 72 (2) ◽  
pp. 455-479
Author(s):  
Shaoxiong Hou ◽  
Deping Ye
Keyword(s):  
Optimization Problem ◽  
Jensen’S Inequality ◽  
Minkowski Inequality ◽  
Isoperimetric Inequalities ◽  
Jensen's Inequality ◽  
Dual Functional ◽  
Two Measures ◽  
Star Bodies

AbstractThis paper provides a functional analogue of the recently initiated dual Orlicz–Brunn–Minkowski theory for star bodies. We first propose the Orlicz addition of measures, and establish the dual functional Orlicz–Brunn–Minkowski inequality. Based on a family of linear Orlicz additions of two measures, we provide an interpretation for the famous $f$-divergence. Jensen’s inequality for integrals is also proved to be equivalent to the newly established dual functional Orlicz–Brunn–Minkowski inequality. An optimization problem for the $f$-divergence is proposed, and related functional affine isoperimetric inequalities are established.

scholarly journals The Dual Orlicz–Aleksandrov–Fenchel Inequality

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2005
Author(s):  
Chang-Jian Zhao
Keyword(s):  
Orlicz Space ◽  
Minkowski Inequality ◽  
Mixed Volume ◽  
Mixed Volumes ◽  
Geometric Quantity ◽  
First Order ◽  
Special Cases ◽  
Star Bodies

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.

scholarly journals Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms

2020 ◽  
Vol 18 (1) ◽  
pp. 1064-1075
Author(s):  
Xia Zhao ◽  
Weidong Wang ◽  
Youjiang Lin
Keyword(s):  
Negative Answer ◽  
Positive Form ◽  
Dual Quermassintegrals ◽  
Monotonic Inequality ◽  
Star Bodies

Abstract In 2006, Schuster introduced the radial Blaschke-Minkowski homomorphisms. In this article, associating with the star duality of star bodies and dual quermassintegrals, we establish Brunn-Minkowski inequalities and monotonic inequality for the radial Blaschke-Minkowski homomorphisms. In addition, we consider its Shephard-type problems and give a positive form and a negative answer, respectively.

scholarly journals New Generalizations and Results in Shift-Invariant Subspaces of Mixed-Norm Lebesgue Spaces \({L_{\vec{p}}(\mathbb{R}^d)}\)

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 227 ◽  
Author(s):  
Junjian Zhao ◽  
Wei-Shih Du ◽  
Yasong Chen
Keyword(s):  
Invariant Subspace ◽  
Type Inequality ◽  
Invariant Subspaces ◽  
Minkowski Inequality ◽  
Hölder Inequality ◽  
Stability Theorem ◽  
Mixed Norm ◽  
Lebesgue Spaces

In this paper, we establish new generalizations and results in shift-invariant subspaces of mixed-norm Lebesgue spaces Lp→(Rd). We obtain a mixed-norm Hölder inequality, a mixed-norm Minkowski inequality, a mixed-norm convolution inequality, a convolution-Hölder type inequality and a stability theorem to mixed-norm case in the setting of shift-invariant subspace of Lp→(Rd). Our new results unify and refine the existing results in the literature.

scholarly journals KHINTCHINE'S THEOREM AND TRANSFERENCE PRINCIPLE FOR STAR BODIES

2006 ◽  
Vol 02 (03) ◽  
pp. 431-453
Author(s):  
M. M. DODSON ◽  
S. KRISTENSEN
Keyword(s):  
Distance Function ◽  
Diophantine Approximation ◽  
Direct Proof ◽  
Distance Functions ◽  
Transference Principle ◽  
Simultaneous Diophantine Approximation ◽  
Star Bodies

Analogues of Khintchine's Theorem in simultaneous Diophantine approximation in the plane are proved with the classical height replaced by fairly general planar distance functions or equivalently star bodies. Khintchine's transference principle is discussed for distance functions and a direct proof for the multiplicative version is given. A transference principle is also established for a different distance function.

On the Dual p-Measures of Asymmetry for Star Bodies

2018 ◽  
Vol 23 (6) ◽  
pp. 465-470
Author(s):  
Xing Huang ◽  
Huawei Zhu ◽  
Qi Guo
Keyword(s):  
Measures Of Asymmetry ◽  
Star Bodies

Prescribing Centro-Affine Curvature From One Convex Body to Another

2020 ◽  
Author(s):  
Alina Stancu
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
Minkowski Inequality ◽  
Curvature Flow ◽  
Constant Volume ◽  
Convex Hypersurface ◽  
Strictly Convex ◽  
Initial Hypersurface ◽  
Affine Curvature ◽  
Convex Hypersurfaces

Abstract We study a curvature flow on smooth, closed, strictly convex hypersurfaces in $\mathbb{R}^n$, which commutes with the action of $SL(n)$. The flow shrinks the initial hypersurface to a point that, if rescaled to enclose a domain of constant volume, is a smooth, closed, strictly convex hypersurface in $\mathbb{R}^n$ with centro-affine curvature proportional, but not always equal, to the centro-affine curvature of a fixed hypersurface. We outline some consequences of this result for the geometry of convex bodies and the logarithmic Minkowski inequality.

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