Brunn-Minkowski Type Inequalities Research Articles

A remark on discrete Brunn–Minkowski type inequalities via transportation of measure

A remark on discrete Brunn–Minkowski type inequalities via transportation of measure

Horocyclic Brunn-Minkowski inequality

Given two non-empty subsets A and B of the hyperbolic plane H2, we define their horocyclic Minkowski sum with parameter λ=1/2 as the set [A:B]1/2⊆H2 of all midpoints of horocycle curves connecting a point in A with a point in B. These horocycle curves are parameterized by hyperbolic arclength. The horocyclic Minkowski sum with parameter 0<λ<1 is defined analogously. We prove that when A and B are Borel-measurable,Area([A:B]λ)≥(1−λ)⋅Area(A)+λ⋅Area(B), where Area stands for hyperbolic area, with equality when A and B are concentric discs in the hyperbolic plane. We also give horocyclic versions of the Prékopa-Leindler and Borell-Brascamp-Lieb inequalities. These inequalities slightly deviate from the metric measure space paradigm on curvature and Brunn-Minkowski type inequalities, where the structure of a metric space is imposed on the manifold, and the relevant curves are necessarily geodesics parameterized by arclength.

A Brunn–Minkowski type inequality for extended symplectic capacities of convex domains and length estimate for a class of billiard trajectories

In this paper, we firstly generalize the Brunn-Minkowski type inequality for Ekeland-Hofer-Zehnder symplectic capacity of bounded convex domains established by Artstein-Avidan-Ostrover in 2008 to extended symplectic capacities of bounded convex domains constructed by authors based on a class of Hamiltonian non-periodic boundary value problems recently. Then we introduce a class of non-periodic billiards in convex domains, and for them we prove some corresponding results to those for periodic billiards in convex domains obtained by Artstein-Avidan-Ostrover in 2012.

Lp-moment mixed quermassintegrals

Lp-moment mixed quermassintegrals of convex bodies in Rn are introduced. The Brunn-Minkowski type inequality and Aleksandrov-Fenchel type inequality are established for Lp-moment mixed quermassintegrals that imply affine mixed quermassintegrals inequality, Lutwak?s mixed polar projection inequality, and isoperimetric inequality for Lp-moment mixed quermassintegrals. Inequalities of Lp-moment mixed quermassintegrals of polar bodies are proved.

On discrete L_p Brunn–Minkowski type inequalities

L_p Brunn–Minkowski type inequalities for the lattice point enumerator mathrm {G}_n(cdot ) are shown, pge 1, both in a geometrical and in a functional setting. In particular, we prove that Gn((1-λ)·K+pλ·L+(-1,1)n)p/n≥(1-λ)Gn(K)p/n+λGn(L)p/n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned}\\mathrm {G}_n\\bigl ((1-\\lambda )\\cdot K +_p \\lambda \\cdot L + (-1,1)^n\\bigr )^{p/n}\\ge (1-\\lambda )\\mathrm {G}_n(K)^{p/n}+\\lambda \\mathrm {G}_n(L)^{p/n} \\end{aligned}$$\\end{document}for any K, Lsubset mathbb {R}^n bounded sets with integer points and all lambda in (0,1). We also show that these new discrete analogues (for mathrm {G}_n(cdot )) imply the corresponding results concerning the Lebesgue measure.

Some Inequalities about the General Lp-Mixed Width-Integral of Convex Bodies

The Brunn-Minkowski type and the cyclic Brunn-Minkowski type inequalities for the i-th general Lp-mixed width-integral of convex bodies are established. Further, two cyclic inequalities for the differences of i-th general Lp-mixed width-integral of convex bodies are obtained.

Inequalities on ComplexLpCentroid Bodies

Based on the notion of the complexLpcentroid body, we establish Brunn-Minkowski type inequalities and monotonicity inequalities for complexLpcentroid bodies in this article. Moreover, we obtain the affirmative form of Shephard type problem for the complexLpcentroid bodies and its negative form.

Discrete variants of Brunn–Minkowski type inequalities

We present an alternative, short proof of a recent discrete version of the Brunn–Minkowski inequality due to Lehec and the second named author. Our proof also yields the four functions theorem of Ahlswede and Daykin as well as some new variants.

Lp -dual three mixed quermassintegrals

Abstract In the paper, the concept of Lp -dual three-mixed quermassintegrals is introduced. The formula for the Lp -dual three-mixed quermassintegrals with respect to the p-radial addition is proved. Inequalities of Lp -Minkowski, and Brunn-Minkowski type for the Lp -dual three-mixed quermassintegrals are established. The new Lp -Minkowski inequality is obtained that generalize a family of Minkowski type inequalities. The Lp -Brunn-Minkowski inequality is used to obtain a series of Brunn-Minkowski type inequalities.

The sharper form of a Brunn-Minkowski type inequality for boxes

In this study, the Brunn-Minkowski inequality for boxes is studied and a sharper version of this inequality is derived by performing the results based on abstract convexity.

Interpolation of Weighted Extremal Functions

An approach to interpolation of compact subsets of {{mathbb {C}}}^n, including Brunn–Minkowski type inequalities for the capacities of the interpolating sets, was developed in [8] by means of plurisubharmonic geodesics between relative extremal functions of the given sets. Here we show that a much better control can be achieved by means of the geodesics between weighted relative extremal functions. In particular, we establish convexity properties of the capacities that are stronger than those given by the Brunn–Minkowski inequalities.

Some inequalities for the (p,q)-mixed geominimal surface areas and Lp radial Blaschke-Minkowski homomorphisms

Wang et al. introduced Lp radial Blaschke-Minkowski homomorphisms based on Schuster?s radial Blaschke-Minkowski homomorphisms. In 2018, Feng and He gave the concept of (p,q)-mixed geominimal surface area according to the Lutwak, Yang and Zhang?s (p,q)-mixed volume. In this article, associated with the (p,q)-mixed geominimal surface areas and the Lp radial Blaschke-Minkowski homomorphisms, we establish some inequalities including two Brunn-Minkowski type inequalities, a cyclic inequality and two monotonic inequalities.

Brunn-Minkowski type inequalities for the lattice point enumerator

Geometric and functional Brunn-Minkowski type inequalities for the lattice point enumerator Gn(⋅) are provided. In particular, we show thatGn((1−λ)K+λL+(−1,1)n)1/n≥(1−λ)Gn(K)1/n+λGn(L)1/n for any non-empty bounded sets K,L⊂Rn and all λ∈(0,1).We also show that these new discrete versions imply the classical results, and discuss some links with other related inequalities.

On Brunn–Minkowski Type Inequalities and Overdetermined Problem for p-Capacity

In this paper, we mainly study some properties of p-capacity. Firstly, the p-capacitary difference Brunn–Minkowski and Minkowski inequalities are established. Secondly, we propose the p-capacitary overdetermined problem and prove the Serrin type symmetry result for the problem. Finally, we also make some considerations for the polar set of p-difference body for capacity.

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.

Representation formula for symmetrical symplectic capacity and applications

This is the second installment in a series of papers aimed at generalizing symplectic capacities and homologies. We study symmetric versions of symplectic capacities for real symplectic manifolds, and obtain corresponding results for them to those of the first [ 19 ] of this series (such as representation formula, a theorem by Evgeni Neduv, Brunn-Minkowski type inequality and Minkowski billiard trajectories proposed by Artstein-Avidan-Ostrover).

New Brunn-Minkowski Type Inequalities for General Width-Integral of Index i

Recently, the general width-integral of index i was introduced and some of its isoperimetric inequalities were established. In this paper, we establish some new Brunn-Minkowski type inequalities for general width-integral of index i.

The Brunn–Minkowski inequality for the principal eigenvalue of fully nonlinear homogeneous elliptic operators

We prove that the principal eigenvalue of any fully nonlinear homogeneous elliptic operator which fulfills a very simple convexity assumption satisfies a Brunn-Minkowski type inequality on the class of open bounded sets in Rn satisfying a uniform exterior sphere condition. In particular the result applies to the (possibly normalized) p-Laplacian, and to the minimal Pucci operator. The proof is inspired by the approach introduced by Colesanti for the principal frequency of the Laplacian within the class of convex domains, and relies on a generalization of the convex envelope method by Alvarez-Lasry-Lions. We also deal with the existence and log-concavity of positive viscosity eigenfunctions.

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.

Lp-dual affine surface areas for the general Lp-intersection bodies

For 0 &lt; p &lt; 1, the notions of symmetric and asymmetric Lp-intersection bodies were introduced by Haberl and Ludwig. Recently, Wang and Li defined the general Lp-intersection bodies. In this paper, associated with the Lp-dual affine surface areas, we give the extremum values of the general Lp-intersection bodies. Moreover, a Brunn-Minkowski type inequality and a monotone inequality for the Lp-dual affine surface area version of general Lp-intersection bodies are established, respectively.