Orlicz Brunn Minkowski Theory Research Articles

SL(n) contravariant function-valued valuations on polytopes

We present a complete classification of SL(n) contravariant, C(Rn∖{o})-valued valuations on polytopes, without any additional assumptions. It extends the previous results of the second author Li (2020) [10] which have a good connection with the Lp and Orlicz Brunn-Minkowski theory. Additionally, our results deduce a complete classification of SL(n) contravariant symmetric-tensor-valued valuations on polytopes.

Orlicz Brunn-Minkowski theory

In this note, we mainly introduce the Orlicz Brunn-Minkowski theory from three aspects:Orlicz projection bodies and Orlicz centroid bodies,Orlicz addition and related volume inequalities, and Orlicz Minkowski problems.

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

The general dual volume $\dveV(K)$ and the general dual Orlicz curvature measure $\deV(K, \cdot)$ were recently introduced for functions $G: (0, \infty)\times \sphere\rightarrow (0, \infty)$ and convex bodies $K$ in $\R^n$ containing the origin in their interiors. We extend $\dveV(K)$ and $\deV(K, \cdot)$ to more general functions $G: [0, \infty)\times \sphere\rightarrow [0, \infty)$ and to compact convex sets $K$ containing the origin (but not necessarily in their interiors). Some basic properties of the general dual volume and of the dual Orlicz curvature measure, such as the continuous dependence on the underlying set, are provided. These are required to study a Minkowski-type problem for the dual Orlicz curvature measure. We mainly focus on the case when $G$ and $\psi$ are both increasing, thus complementing our previous work. The Minkowski problem asks to characterize Borel measures $\mu$ on $\sphere$ for which there is a convex body $K$ in $\R^n$ containing the origin such that $\mu$ equals $\deV(K, \cdot)$, up to a constant. A major step in the analysis concerns discrete measures $\mu$, for which we prove the existence of convex polytopes containing the origin in their interiors solving the Minkowski problem. For general (not necessarily discrete) measures $\mu$, we use an approximation argument. This approach is also applied to the case where $G$ is decreasing and $\psi$ is increasing, and hence augments our previous work. When the measures $\mu$ are even, solutions that are origin-symmetric convex bodies are also provided under some mild conditions on $G$ and $\psi$. Our results generalize several previous works and provide more precise information about the solutions of the Minkowski problem when $\mu$ is discrete or even.

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

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.

The Orlicz Brunn-Minkowski Inequality for Dual Harmonic Quermassintegrals

Within the framework of Orlicz Brunn-Minkowski theory recently introduced by Lutwak, Yang, and Zhang [20, 21], Gardner, Hug, and Weil [5, 6] et al, the dual harmonic quermassintegrals of star bodies are studied, and a new Orlicz Brunn-Minkowski type inequality is proved for these geometric quantities.

The Orlicz Brunn–Minkowski Inequality for the Projection Body

A generalized Brunn–Minkowski inequality for the projection body is established in the framework of the Orlicz Brunn–Minkowski theory. This new inequality yields the Orlicz Brunn–Minkowski inequality for the intrinsic volume directly.

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.

Affine Function-Valued Valuations

A classification of SL$(n)$ contravariant, continuous function valued valuations on convex bodies is established. Such valuations are natural extensions of SL$(n)$ contravariant $L_p$ Minkowski valuations, the classification of which characterized $L_p$ projection bodies, which are fundamental in the $L_p$ Brunn-Minkowski theory, for $p \geq 1$. Hence our result will help to better understand extensions of the $L_p$ Brunn-Minkowski theory. In fact, our results characterize general projection functions which extend $L_p$ projection functions ($p$-th powers of the support functions of $L_p$ projection bodies) to projection functions in the $L_p$ Brunn-Minkowski theory for $0< p < 1$ and in the Orlicz Brunn-Minkowski theory.

The p-capacitary Orlicz–Hadamard variational formula and Orlicz–Minkowski problems

In this paper, combining the p-capacity for $$p\in (1, n)$$ with the Orlicz addition of convex domains, we develop the p-capacitary Orlicz–Brunn–Minkowski theory. In particular, the Orlicz $$L_{\phi }$$ mixed p-capacity of two convex domains is introduced and its geometric interpretation is obtained by the p-capacitary Orlicz–Hadamard variational formula. The p-capacitary Orlicz–Brunn–Minkowski and Orlicz–Minkowski inequalities are established, and the equivalence of these two inequalities are discussed as well. The p-capacitary Orlicz–Minkowski problem is proposed and solved under some mild conditions on the involving functions and measures. In particular, we provide the solutions for the normalized p-capacitary $$L_q$$ Minkowski problems with $$q>1$$ for both discrete and general measures.

Affine isoperimetric inequalities in the functional Orlicz–Brunn–Minkowski theory

In this paper, we develop a basic theory of Orlicz affine and geominimal surface areas for convex and s-concave functions. We prove some basic properties for these newly introduced functional affine invariants and establish related functional affine isoperimetric inequalities as well as functional Santaló type inequalities.

Dual Orlicz–Brunn–Minkowski theory: Dual Orlicz Lϕ affine and geominimal surface areas

This paper aims to develop the basic theory for the dual Orlicz Lϕ affine and geominimal surface areas of star bodies. Basic properties of these new affine invariants are established. Moreover, related Orlicz affine isoperimetric, cyclic, Santaló type and Alexandrov–Fenchel type inequalities are obtained.

Orlicz–Legendre Ellipsoids

The Orlicz–Legendre ellipsoids, which are in the framework of emerging dual Orlicz Brunn–Minkowski theory, are introduced for the first time. They are in some sense dual to the recently found Orlicz–John ellipsoids, and have largely generalized the classical Legendre ellipsoid of inertia. Several new affine isoperimetric inequalities are established. The connection between the characterization of Orlicz–Legendre ellipsoids and isotropy of measures is demonstrated.

The dual Orlicz–Brunn–Minkowski theory

A first step towards a dual Orlicz–Brunn–Minkowski theory for star sets was taken by Zhu, Zhou, and Xue [44,45]. In this essentially independent work we provide a more general framework and results. 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 Böröczky, 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.

New Orlicz affine isoperimetric inequalities

The Orlicz–Brunn–Minkowski theory received considerable attention recently, and many results in the Lp-Brunn–Minkowski theory have been extended to their Orlicz counterparts. The aim of this paper is to develop Orlicz Lϕ affine and geominimal surface areas for a single convex body as well as for multiple convex bodies, which generalize the Lp (mixed) affine and geominimal surface areas – fundamental concepts in the Lp-Brunn–Minkowski theory. Our extensions are different from the general affine surface areas by Ludwig in [21]. Moreover, our definitions for Orlicz Lϕ affine and geominimal surface areas reveal that these affine invariants are essentially the infimum/supremum of Vϕ(K,L∘), the Orlicz ϕ-mixed volume of K and the polar body of L, where L runs over all star bodies and all convex bodies, respectively, with volume of L equal to the volume of the unit Euclidean ball B2n. Properties for the Orlicz Lϕ affine and geominimal surface areas, such as affine invariance and monotonicity, are proved. Related Orlicz affine isoperimetric inequalities are also established.

Orlicz mixed affine quermassintegrals

A class of geometric quantities for convex bodies is introduced in the framework of Orlicz Brunn-Minkowski theory. It is shown that these new geometric quantities are affine invariant and precisely the generalizations of classical affine quermassintegrals.

Orlicz–John ellipsoids

The Orlicz–John ellipsoids, which are in the framework of the booming Orlicz Brunn–Minkowski theory, are introduced for the first time. It turns out that they are generalizations of the classical John ellipsoid and the evolved Lp John ellipsoids. The analog of Ball's volume-ratio inequality is established for the new Orlicz–John ellipsoids. The connection between the isotropy of measures and the characterization of Orlicz–John ellipsoids is demonstrated.

Dual Orlicz–Brunn–Minkowski theory

In this paper, a dual Orlicz–Brunn–Minkowski theory is presented. An Orlicz radial sum and dual Orlicz mixed volumes are introduced. The dual Orlicz–Minkowski inequality and the dual Orlicz–Brunn–Minkowski inequality are established. The variational formula for the volume with respect to the Orlicz radial sum is proved. The equivalence between the dual Orlicz–Minkowski inequality and the dual Orlicz–Brunn–Minkowski inequality is demonstrated. Orlicz intersection bodies are defined and the Orlicz–Busemann–Petty problem is posed.

The Orlicz Brunn–Minkowski inequality

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 Lp Brunn–Minkowski inequality to the Orlicz Brunn–Minkowski inequality. Furthermore, we extend the Lp Minkowski mixed volume inequality to the Orlicz mixed volume inequality by using the Orlicz Brunn–Minkowski inequality.

Orlicz mixed quermassintegrals

The notion of mixed quermassintegrals in the classical Brunn-Minkowski theory is extended to that of Orlicz mixed quermassintegrals in the Orlicz Brunn-Minkowski theory. The analogs of the classical Cauchy-Kubota formula, the Minkowski isoperimetric inequality and the Brunn-Minkowski inequality are established for this new Orlicz mixed quermassintegrals.