Dual Mixed Volumes Research Articles

Dual Brunn-Minkowski inequality for $ C $-star bodies

<abstract><p>In this paper, we introduced the concept of $ C $-star bodies in a fixed pointed closed convex cone $ C $ and studied the dual mixed volume for $ C $-star bodies. For $ C $-star bodies, we established the corresponding dual Brunn-Minkowski inequality, dual Minkowski inequality, and dual Aleksandrov-Fenchel inequality. Moveover, we found that the dual Brunn-Minkowski inequality for $ C $-star bodies can strengthen the Brunn-Minkowski inequality for $ C $-coconvex sets.</p></abstract>

Lp-Steiner quermassintegrals

Inspired by an Lp Steiner formula for the Lp affine surface area proved by Tatarko and Werner, we define, in analogy to the classical Steiner formula, Lp-Steiner quermassintegrals. Special cases include the classical mixed volumes, the dual mixed volumes, the Lp affine surface areas and the mixed Lp affine surface areas. We investigate the properties of the Lp-Steiner quermassintegrals in a special class of convex bodies. In particular, we show that they are rotation and reflection invariant valuations in this class of convex bodies with a certain degree of homogeneity. Such valuations seem new and have not been observed before.

Orlicz dual of log-Aleksandrov–Fenchel inequality

In this paper, we establish an Orlicz dual of the log-Aleksandrov–Fenchel inequality, by introducing two new concepts of dual mixed volume measures, and using the newly established Orlicz dual Aleksandrov–Fenchel inequality. The Orlicz dual log-Aleksandrov– Fenchel inequality in special cases yields the classical dual Aleksandrov–Fenchel inequality and some dual logarithmic Minkowski type inequalities, respectively. Moreover, the dual log-Aleksandrov–Fenchel inequality is therefore also derived.

The functional form of the dual mixed volume

This paper is to generalize the dual mixed volume of star bodies to that of dual quasi-concave functions by extending the radial Minkowski linear combination of star bodies to that of dual quasi-concave functions. We attempt to build up some functional versions of notions and inequalities from the dual Brunn-Minkowski theory. In particular, both the dual mixed Brunn-Minkowski inequality of star bodies and the dual Aleksandrov Fenchel inequality of star bodies are generalized to that of dual quasi-concave functions.

Cyclic Brunn–Minkowski inequalities for dual mixed volumes

Combined with radial Minkowski addition, radial Blaschke addition and harmonic Blaschke addition, we establish cyclic Brunn–Minkowski inequalities for dual mixed volumes of star bodies.

The Dual Orlicz–Aleksandrov–Fenchel Inequality

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.

A Functional Busemann Intersection Inequality

The Busemann intersection inequality states that if K is a compact domain in $${\mathbb {R}}^n$$ then where $$c(n)>0$$ is an explicit constant, with equality if and only if K is an ellipsoid centered at the origin. In this paper, we prove a functional version of the Busemann intersection inequality. We also demonstrate an “equivalent” sharp entropy inequality for dual mixed volumes of functions.

The General Dual Orlicz Geominimal Surface Area

In this paper, we study the general dual Orlicz geominimal surface area by the general dual Orlicz mixed volume which was introduced by Gardner et al. (2019). We find the conditions to the existence of the general dual Orlicz-Petty body and hence prove the continuity of the general geominimal surface area in the Orlicz setting (2010 Mathematics Subject Classification: 52A20, 53A15).

Inequalities for the Mixed Radial Blaschke-Minkowski Homomorphisms and the Applications

The notion of intersection body is introduced by Lutwak in 1988, it is one of important research contents and led to the studies of Busemann-Petty problem in the Brunn-Minkowski theory. Based on the properties of the intersection bodies, Schuster introduced the notion of radial Blaschke-Minkowski homomorphisms and proved a lot of related inequalities. In this paper, by applying the dual mixed volume theory and analytic inequalities, we first give a lower bound of the dual quermassintegrals for the mixed radial Blaschke-Minkowski homomorphisms. As its an application, we get a reverse form of the well-known Busemann intersection inequality. Further, a Brunn-Minkowski type inequality of the Lp radial Minkowski sum for the dual quermassintegrals of mixed radial Blaschke-Minkowski homomorphisms is established, and then the intersection body version of this Brunn-Minkowski type inequality is yielded. From this, we not only extend Schuster's related result but also obtain the Brunn-Minkowski type inequalities of Lp harmonic radial sum and Lp radial Blaschke sum, respectively.

General volumes in the Orlicz–Brunn–Minkowski theory and a related Minkowski problem I

The general volume of a star body, a notion that includes the usual volume, the $q$th dual volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new general dual Orlicz curvature measure is defined that specializes to the $(p,q)$-dual curvature measures introduced recently by Lutwak, Yang, and Zhang. General variational formulas are established for the general volume of two types of Orlicz linear combinations. One of these is applied to the Minkowski problem for the new general dual Orlicz curvature measure, giving in particular a solution to the Minkowski problem posed by Lutwak, Yang, and Zhang for the $(p,q)$-dual curvature measures when $p>0$ and $q<0$. A dual Orlicz-Brunn-Minkowski inequality for general volumes is obtained, as well as dual Orlicz-Minkowski-type inequalities and uniqueness results for star bodies. Finally, a very general Minkowski-type inequality, involving two Orlicz functions, two convex bodies, and a star body, is proved, that includes as special cases several others in the literature, in particular one due to Lutwak, Yang, and Zhang for the $(p,q)$-mixed volume.

On the Dual p-Measures of Asymmetry for Star Bodies

Recently, the connection between p-measures of asymmetry and the Lp -mixed volumes for convex bodies was found soon after the p-measure of asymmetry was proposed, and the Orlicz-measures of asymmetry was proposed inspired by such a kind of connection. In this paper, by a similar way the dual p -measures of asymmetry for star bodies (naturally for convex bodies) is introduced first. Then the connection between dual p -measures of asymmetry and Lp -dual mixed volumes is established. Finally, the best lower and upper bounds of dual p-measures and the corresponding extremal bodies are discussed.

Some inequalities for (p,q)-mixed volume

Lutwak, Yang, and Zhang introduced the concept of (p,q)-mixed volume whose special cases contain the L_{p}-mixed volume and the L_{p}-dual mixed volume. In this article, associated with the (p,q)-mixed volumes, we establish related cyclic inequalities, monotonic inequalities, and product inequalities.

Lp dual curvature measures

A new family of geometric Borel measures on the unit sphere is introduced. Special cases include the Lp surface area measures (which extend the classical surface area measure of Aleksandrov and Fenchel & Jessen) and Lp-integral curvature (which extends Alkesandrov's integral curvature) in the Lp Brunn–Minkowski theory. It also includes the dual curvature measures (which are the duals of Federer's curvature measures) in the dual Brunn–Minkowski theory. This partially unifies the classical theory of mixed volumes and the newer theory of dual mixed volumes.

Orlicz Mean Dual Affine Quermassintegrals

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

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 {L_{p}} -dual Minkowski inequality and Brunn–Minkowski inequality for the dual affine quermassintegrals.

Dual Orlicz Mixed Affine Quermassintegrals

In this paper, a class of geometric quantities of star bodies via dual Orlicz mixed volumes are presented. It is proved that these geometric quantities are affine invariant and precisely the generalizations of their classical counterparts. Further, some related inequalities with respect to these geometric quantities are established.

Q-dual mixed volumes and L p -intersection bodies

Abstract In this paper, we introduce the new notions of L p {L_{p}} -intersection and mixed intersection bodies. Inequalities for the q-dual volume sum of L p {L_{p}} -mixed intersection bodies are established.

The dual difference Aleksandrov–Fenchel inequality

Recently, a dual Minkowski inequality and a dual Brunn–Minkowski inequality for volume differences were established. Following this, in this paper we establish a dual Aleksandrov–Fenchel inequality for dual mixed volume differences which generalizes several recent results.

Dual Orlicz geominimal surface area

The $L_{p}$ -geominimal surface area was introduced by Lutwak in 1996, which extended the important concept of the geominimal surface area. Recently, Wang and Qi defined the p-dual geominimal surface area, which belongs to the dual Brunn-Minkowski theory. In this paper, based on the concept of the dual Orlicz mixed volume, we extend the dual geominimal surface area to the Orlicz version and give its properties. In addition, the isoperimetric inequality, a Blaschke-Santalo type inequality, and the monotonicity inequality for the dual Orlicz geominimal surface areas are established.

L_p-mixed intersection bodies

In this paper, we prove a dual Bergstrom type inequality for q -dual mixed volumes. Furthermore, we introduce the Lp -mixed intersection bodies for any real number p = 0 , and establish some interesting inequalities for Lp -mixed intersection bodies involving a Minkowski type inequality and three Brunn-Minkowski type inequalities. Mathematics subject classification (2010): 52A38, 52A40.