scholarly journals Orlicz Mean Dual Affine Quermassintegrals

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Chang-Jian Zhao ◽  
Wing-Sum Cheung
Keyword(s):  
Orlicz Space ◽  
Minkowski Inequality ◽  
Isoperimetric Inequalities ◽  
Mixed Volumes ◽  
Geometric Quantity ◽  
Dual Affine ◽  
The Mean ◽  
Dual Minkowski Inequality ◽  
Special Case

Our main aim is to generalize the mean dual affine quermassintegrals to the Orlicz space. Under the framework of dual Orlicz-Brunn-Minkowski theory, we introduce a new affine geometric quantity by calculating the first Orlicz variation of the mean dual affine quermassintegrals and call it the Orlicz mean dual affine quermassintegral. The fundamental notions and conclusions of the mean dual affine quermassintegrals and the Minkowski and Brunn-Minkowski inequalities for them are extended to an Orlicz setting. The related concepts and inequalities of dual Orlicz mixed volumes are also included in our conclusions. The new Orlicz isoperimetric inequalities in special case yield theLp-dual Minkowski inequality and Brunn-Minkowski inequality for the mean dual affine quermassintegrals, which also imply the dual Orlicz-Minkowski inequality and dual Orlicz-Brunn-Minkowski inequality.

Orlicz dual affine quermassintegrals

2018 ◽  
Vol 30 (4) ◽  
pp. 929-945 ◽  
Author(s):  
Chang-Jian Zhao
Keyword(s):  
Orlicz Space ◽  
Minkowski Inequality ◽  
Mixed Volumes ◽  
Geometric Quantity ◽  
First Order ◽  
Order Variation ◽  
Dual Affine ◽  
Dual Minkowski Inequality ◽  
Dual Mixed Volumes ◽  
Special Case

Abstract In the paper, our main aim is to generalize the dual affine quermassintegrals to the Orlicz space. Under the framework of Orlicz dual Brunn–Minkowski theory, we introduce a new affine geometric quantity by calculating the first-order variation of the dual affine quermassintegrals, and call it the Orlicz dual affine quermassintegral. The fundamental notions and conclusions of the dual affine quermassintegrals and the Minkoswki and Brunn–Minkowski inequalities for them are extended to an Orlicz setting, and the related concepts and inequalities of Orlicz dual mixed volumes are also included in our conclusions. The new Orlicz–Minkowski and Orlicz–Brunn–Minkowski inequalities in a special case yield the Orlicz dual Minkowski inequality and Orlicz dual Brunn–Minkowski inequality, which also imply the {L_{p}} -dual Minkowski inequality and Brunn–Minkowski inequality for the dual affine quermassintegrals.

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 Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes

2018 ◽  
Vol 2018 ◽  
pp. 1-16 ◽  
Author(s):  
Chang-Jian Zhao
Keyword(s):  
Orlicz Space ◽  
Convex Bodies ◽  
Minkowski Inequality ◽  
Mixed Volume ◽  
Mixed Volumes ◽  
Geometric Quantity ◽  
First Order ◽  
Special Cases ◽  
Order Variation

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.

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.

Portfolio selection and matching: a synthesis

1992 ◽  
Vol 119 (1) ◽  
pp. 87-105 ◽  
Author(s):  
M. Sherris
Keyword(s):  
Portfolio Selection ◽  
Life Insurance ◽  
General Framework ◽  
Pension Fund ◽  
Central Value ◽  
Efficient Portfolios ◽  
The Mean ◽  
Mean Variance ◽  
Special Case ◽  
Selection Of

AbstractThis paper considers a general framework for the selection of assets to meet the liabilities of a life insurance or pension fund. This general framework contains the mean-variance efficient portfolios of modern portfolio theory as a special case. The paper also demonstrates how the portfolio selection and matching approach of Wise (1984a, 1984b, 1987a, 1987b) and Wilkie (1985) fits into this general framework. The matching portfolio is derived as a special case, and is also shown to have implications for determining the central value of the liabilities.

scholarly journals A note on the ruled surface

1937 ◽  
Vol 30 ◽  
pp. i-ii
Author(s):  
R. Wilson
Keyword(s):  
Mean Curvature ◽  
Geodesic Curvature ◽  
Ruled Surface ◽  
Local Geometry ◽  
Parametric Curves ◽  
The Mean ◽  
Curvature And Torsion ◽  
Asymptotic Lines ◽  
Special Case ◽  
Elegant Form

The generators and their orthogonal trajectories form, perhaps, the most useful set of parametric curves for the study of the local geometry of a ruled surface. It is not generally realised, however, that the fundamental quantities of the surface can be expressed quite simply in terms of the geodesic curvature, the geodesic torsion and the normal curvature of the directrix, that particular orthogonal trajectory which is chosen as base curve. Certain of the results are similar in form to those arising in the special case of a surface which is generated by the principal normals to a given curve, except that the curvature and torsion are geodetic. In addition it is possible to obtain in an elegant form the differential equation of the curved asymptotic lines and the expression for the mean curvature.

The Largest Missing Value in a Sample of Geometric Random Variables

2014 ◽  
Vol 23 (5) ◽  
pp. 670-685 ◽  
Author(s):  
MARGARET ARCHIBALD ◽  
ARNOLD KNOPFMACHER
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Missing Value ◽  
Asymptotic Expressions ◽  
Mean And Variance ◽  
The Mean ◽  
Special Case

We consider samples of n geometric random variables with parameter 0 < p < 1, and study the largest missing value, that is, the highest value of such a random variable, less than the maximum, that does not appear in the sample. Asymptotic expressions for the mean and variance for this quantity are presented. We also consider samples with the property that the largest missing value and the largest value which does appear differ by exactly one, and call this the LMV property. We find the probability that a sample of n variables has the LMV property, as well as the mean for the average largest value in samples with this property. The simpler special case of p = 1/2 has previously been studied, and verifying that the results of the present paper coincide with those previously found for p = 1/2 leads to some interesting identities.

Unsteady plane flow past curved obstacles with infinite wakes

1955 ◽  
Vol 229 (1177) ◽  
pp. 152-180 ◽  
Keyword(s):  
Unsteady Flow ◽  
Classical Theory ◽  
Steady Motion ◽  
Uniform Motion ◽  
Plane Flow ◽  
Flow Past ◽  
Long Duration ◽  
Infinite Extent ◽  
The Mean ◽  
Special Case

A theory of unsteady flow about obstacles behind which are wakes or cavities of infinite extent is developed for the case when the velocities and displacements of the unsteady perturbations about the mean steady motion are small. Unsteady Helmholtz flows (constant wake pressure) receive detailed attention both for general non-uniform motion and for the special case of harmonic motions of long duration. A number of possible applications of the theory to aerodynamic problems are indicated, the most important being the flutter of a stalled aerofoil. The classical theory of unsteady aerofoik motion is shown to be a special case of the theory given in this paper.

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