Orlicz Brunn Minkowski Inequality Research Articles

The functional Orlicz–Brunn–Minkowski inequality for q‐torsional rigidity

AbstractIn this paper, we obtain the functional Orlicz–Brunn–Minkowski inequality and the functional Orlicz–Minkowski inequality for q‐torsional rigidity in the smooth category. Furthermore, using an approximation method, we give the general functional Orlicz–Brunn–Minkowski inequality for q‐torsional rigidity. As a corollary, we reveal that the functional Orlicz–Brunn–Minkowski inequality is equivalent to the functional Orlicz–Minkowski inequality for q‐torsional rigidity in the smooth category. We also give some applications with respect to these two inequalities.

The Orlicz Brunn-Minkowski inequality for the first eigenvalue of p-Laplacian

In this paper, we obtain the Orlicz Hadamard variational formula and the Orlicz Brunn-Minkowski inequality for the first eigenvalue of p-Laplacian in the smooth case. Furthermore, the uniqueness of the Orlicz mixed first eigenvalue of p-Laplacian with p>1 is proved.

The dual Orlicz Brunn–Minkowski inequality for the intersection body

The dual Orlicz Brunn–Minkowski inequality for the intersection body

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.

The Functional Orlicz Brunn-Minkowski Inequality for q-Capacity

In this paper, we establish functional forms of the Orlicz Brunn-Minkowski inequality and the Orlicz-Minkowski inequality for the electrostatic q-capacity, which generalize previous results by Zou and Xiong. We also show that these two inequalities are equivalent.

Orlicz mixed radial Blaschke–Minkowski homomorphisms

Our main aim is to generalize the mixed radial Blaschke–Minkowski homomorphisms to the Orlicz space. By calculating first-order variation of the harmonic p-dual quermassintegrals of mixed radial Blaschke–Minkowski homomorphisms, we introduce a new affine geometric quantity and call it Orlicz mixed radial Blaschke–Minkowski homomorphisms. The Orlicz–Minkowski inequality for Orlicz mixed radial Blaschke–Minkowski homomorphisms is established. As an application, an Orlicz–Brunn–Minkowski inequality for the mixed radial Blaschke–Minkowski homomorphisms is also established, which in special case yields new $$L_p$$ Brunn–Minkowski inequality for mixed radial Blaschke–Minkowski homomorphisms.

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.

A new approach to the Orlicz Brunn-Minkowski inequality

This paper presents a new approach to the recently established Orlicz Brunn-Minkowski inequality which is due to Xi, Jin and Leng and is stronger than the classical and Lp Brunn-Minkowski inequalities. Our approach is based on the shadow system and does not rely on the Steiner symmetrization.

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.

Dual mixed Orlicz–Brunn–Minkowski inequality and dual Orlicz mixed quermassintegrals

The dual Orlicz–Brunn–Minkowski inequality is extended from volume to dual quermassintegrals. As application, a dual mixed log-Brunn–Minkowski inequality is obtained. Moreover, dual Orlicz mixed quermassintegrals are defined and a dual Orlicz–Minkowski inequality and a dual mixed log-Minkowski inequality are established.

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.

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.