Minkowski Inequality Research Articles (Page 10)

In this paper, we introduce and prove the generalizations of Radon inequality. The proofs in the paper unify and are simpler than those in former work. Meanwhile, we also find mathematical equivalences among the Bernoulli inequality, the weighted AM-GM inequality, the H\"{o}lder inequality, the weighted power mean inequality and the Minkowski inequality. Finally, a series of the applications are shown in this note.

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.

We present some of the classical inequalities in analysis in the context of Archimedean (real or complex) vector lattices and f-algebras. In particular, we prove an identity for sesquilinear maps from the Cartesian square of a vector space to a geometric mean closed Archimedean vector lattice, from which a Cauchy–Schwarz inequality follows. A reformulation of this result for sesquilinear maps with a geometric mean closed semiprime Archimedean f-algebra as codomain is also given. In addition, a sufficient and necessary condition for equality is presented. We also prove a Hölder inequality for weighted geometric mean closed Archimedean Φ-algebras, substantially improving results by K. Boulabiar and M. A. Toumi. As a consequence, a Minkowski inequality for weighted geometric mean closed Archimedean Φ-algebras is obtained.

In this paper, we first introduce the definition of triple Diamond-Alpha integral for functions of three variables. Therefore, we present the Hölder and reverse Hölder inequalities for the triple Diamond-Alpha integral on time scales, and then we obtain some new generalizations of the Hölder and reverse Hölder inequalities for the triple Diamond-Alpha integral. Moreover, using the obtained results, we give a new generalization of the Minkowski inequality for the triple Diamond-Alpha integral on time scales.

Given one metric measure space X satisfying a linear Brunn–Minkowski inequality, and a second one Y satisfying a Brunn–Minkowski inequality with exponent p≥−1, we prove that the product X×Y with the standard product distance and measure satisfies a Brunn–Minkowski inequality of order 1/(1+p−1) under mild conditions on the measures and the assumption that the distances are strictly intrinsic. The same result holds when we consider restricted classes of sets. We also prove that a linear Brunn–Minkowski inequality is obtained in X×Y when Y satisfies a Prékopa–Leindler inequality.In particular, we show that the classical Brunn–Minkowski inequality holds for any pair of weakly unconditional sets in Rn (i.e., those containing the projection of every point in the set onto every coordinate subspace) when we consider the standard distance and the product measure of n one-dimensional real measures with positively decreasing densities. This yields an improvement of the class of sets satisfying the Gaussian Brunn–Minkowski inequality.Furthermore, associated isoperimetric inequalities as well as recently obtained Brunn–Minkowski's inequalities are derived from our results.

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.

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.

New Orlicz Brunn–Minkowski inequalities are established for rigid motion compatible Minkowski valuations of arbitrary degree. These extend classical log-concavity properties of intrinsic volumes and generalize seminal results of Lutwak and others. Two different approaches which refine previously employed techniques are explored. It is shown that both lead to the same class of Minkowski valuations for which these inequalities hold. An appendix by Semyon Alesker contains the proof of a new classification of generalized translation invariant valuations.

One of the famous mathematical inequality is Minkowski's inequality. It is an important inequality from both mathematical and application points of view. In this paper, a Minkowski type inequality for fuzzy and pseudo-integrals is studied. The established results are based on the classical Minkowski's inequality for integrals.

Some integral inequalities for fuzzy-interval-valued functions

Abstract In this paper, we introduce the anisotropic Sobolev capacity with fractional order and develop some basic properties for this new object. Applications to the theory of anisotropic fractional Sobolev spaces are provided. In particular, we give geometric characterizations for a nonnegative Radon measure μ that naturally induces an embedding of the anisotropic fractional Sobolev class into the μ-based-Lebesgue-space with 0 < β ≤ n. Also, we investigate the anisotropic fractional α-perimeter. Such a geometric quantity can be used to approximate the anisotropic Sobolev capacity with fractional order. Estimation on the constant in the related Minkowski inequality, which is asymptotically optimal as α →0+, will be provided.

This paper is devoted to investigating a mixed volume from the anisotropic potential with natural logarithm as a better complement to the end point case of the most recently developed mixed volumes from the anisotropic Riesz-potential. An optimal polynomial \(\log \)-inequality is not only discovered but also applicable to produce a polynomial dual for the conjectured fundamental \(\log \)-Minkowski inequality in convex geometry analysis, whence generalizing the dual \(\log \)-Minkowski inequality for mixed volume of two star bodies.

We prove some analogs inequalities of the logarithmic Minkowski inequality for general nonsymmetric convex bodies. As applications of one of those inequalities, the p-affine isoperimetric inequality and some other inequalities are obtained.

A generalization of Lp-Brunn–Minkowski inequalities and Lp-Minkowski problems for measures

We provide a sharp quantitative version of the Gaussian concentration inequality: for every $r>0$, the difference between the measure of the $r$-enlargement of a given set and the $r$-enlargement of a half-space controls the square of the measure of the symmetric difference between the set and a suitable half-space. We also prove a similar estimate in the Euclidean setting for the enlargement with a general convex set. This is equivalent to the stability of the Brunn-Minkowski inequality for the Minkowski sum between a convex set and a generic one.

Quantitative stability for the Brunn–Minkowski inequality

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

On the stability of Brunn–Minkowski type inequalities

We establish some generalized Hölder’s and Minkowski’s inequalities for Jackson’sq-integral. As applications, we derive some inequalities involving the incompleteq-Gamma function.

The theory of coconvex bodies was formalized by A. Khovanskiĭ and V. Timorin in [4]. It has fascinating relations with the classical theory of convex bodies, as well as applications to Lorentzian geometry. In a recent preprint [5], R. Schneider proved a result that implies a reversed Brunn–Minkowski inequality for coconvex bodies, with description of equality case. In this note we show that this latter result is an immediate consequence of a more general result, namely that the volume of coconvex bodies is strictly convex. This result itself follows from a classical elementary result about the concavity of the volume of convex bodies inscribed in the same cylinder.