Minkowski Inequality Research Articles

This paper aims to consider the dual Brunn–Minkowski inequality for log-volume of star bodies, and the equivalent Minkowski inequality for mixed log-volume.

In this study, we extend some “sneak-out” inequalities on time scales for a function depending on more than one parameter. The results are proved by using the induction principle and time scale version of Minkowski inequalities. In seeking applications, these inequalities are discussed in classical, discrete, and quantum calculus.

More on Minkowski and Hardy integral inequality on time scales

Mixed [formula omitted] projection inequality

In this research, we introduce some new fractional integral inequalities of Minkowski’s type by using Riemann-Liouville fractional integral operator. We replace the constants that appear on Minkowski’s inequality by two positive functions. Further, we establish some new fractional inequalities related to the reverse Minkowski type inequalities via Riemann-Liouville fractional integral. Using this fractional integral operator, some special cases of reverse Minkowski type are also discussed.

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.

Abstract In this paper some properties of the pseudo-integral are summarized and a characterization theorem for this integral is proposed. Using the characterization theorem, we obtain that the pseudo-integral with respect to the pseudo-product of two σ-⊕-measures can be reduced to repeated pseudo-integrals. As a consequence of that claim and the Hölder type inequality for the pseudo-integral, we get the generalized Minkowski inequality for the pseudo-integral.

In this paper, we establish new generalizations and results in shift-invariant subspaces of mixed-norm Lebesgue spaces Lp→(Rd). We obtain a mixed-norm Hölder inequality, a mixed-norm Minkowski inequality, a mixed-norm convolution inequality, a convolution-Hölder type inequality and a stability theorem to mixed-norm case in the setting of shift-invariant subspace of Lp→(Rd). Our new results unify and refine the existing results in the literature.

<abstract> In the paper, our main aim is to generalize the mixed affine quermassintegrals of <italic>j</italic> convex bodies to the Orlicz space. We find a new affine geometric quantity by calculating first-order variation and call it <italic>Orlicz multiple affine quermassintegrals</italic>. The mixed affine quermassintegrals and AleksandrovFenchel inequality for the mixed affine quermassintegrals of <italic>j</italic> convex bodies are extended to an Orlicz setting. A new Orlicz-Aleksandrov-Fenchel inequality for the mixed affine quermassintegrals of <italic>j</italic> convex bodies is established. The new Orlicz-Aleksandrov-Fenchel inequality in special cases yield the classical Aleksandrov-Fenchel inequality for mixed volumes, the Aleksandrov-Fenchel inequality for the mixed affine quermassintegrals which is just built, and Zou's Orlicz Minkowski inequality for affine quermassintegrals, respectively. This new concept of <italic>L_p</italic>-multiple affine quermassintegrals and <italic>L_p</italic>-AleksandrovFenchel inequality for the <italic>L_p</italic>-multiple affine quermassintegrals is also derived. Moreover, the Orlicz multiple mixed volumes and the Orlicz-AleksandrovFenchel inequality for the mixed volumes are also included in our new conclusions. As an application, a new Orlicz-Brunn-Minkowski inequality for the mixed affine quermassintegrals of <italic>j</italic> convex bodies is proved. </abstract>

The dimensional Brunn–Minkowski inequality in Gauss space

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.

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.

Orlicz log-Minkowski inequality

Abstract This paper deals with locally constrained inverse curvature flows in a broad class of Riemannian warped spaces. For a certain class of such flows, we prove long-time existence and smooth convergence to a radial coordinate slice. In the case of two-dimensional surfaces and a suitable speed, these flows enjoy two monotone quantities. In such cases, new Minkowski type inequalities are the consequence. In higher dimensions, we use the inverse mean curvature flow to obtain new Minkowski inequalities when the ambient radial Ricci curvature is constantly negative.

We use the properties of superquadratic functions to produce various improvements and popularizations on time scales of the Hardy form inequalities and their converses. Also, we include various examples and interpretations of the disparities in the literature that exist. In particular, our findings can be seen as refinements of some recent results closely linked to the time-scale inequalities of the classical Hardy, Pólya-Knopp, and Hardy-Hilbert. Some continuous inequalities are derived from the main results as special cases. The essential results will be proved by making use of some algebraic inequalities such as the Minkowski inequality, the refined Jensen inequality, and the Bernoulli inequality on time scales.

In the present paper, we first introduce the concepts of the Lpq-capacity measure and Lp mixed q-capacity and then prove some geometric properties of Lpq-capacity measure and a Lp Minkowski inequality for the q-capacity for any fixed p ⩾ 1 and q &gt; n. As an application of the Lp Minkowski inequality mentioned above, we establish a Hadamard variational formula for the q-capacity under p-sum for any fixed p ⩾ 1 and q &gt; n, which extends results of Akman et al. (Adv. Calc. Var. (in press)). With the Hadamard variational formula, variational method and Lp Minkowski inequality mentioned above, we prove the existence and uniqueness of the solution for the Lp Minkowski problem for the q-capacity which extends some beautiful results of Jerison (1996, Acta Math.176, 1–47), Colesanti et al. (2015, Adv. Math.285, 1511–588), Akman et al. (Mem. Amer. Math. Soc. (in press)) and Akman et al. (Adv. Calc. Var. (in press)). It is worth mentioning that our proof of Hadamard variational formula is based on Lp Minkowski inequality rather than the direct argument which was adopted by Akman (Adv. Calc. Var. (in press)). Moreover, as a consequence of Lp Minkowski inequality for q-capacity, we get an interesting isoperimetric inequality for q-capacity.

The famous Minkowski inequality provides a sharp lower bound for the mixed volume V ( K , M [ n − 1 ] ) V(K,M[n-1]) of two convex bodies K , M ⊂ R n K,M\subset \mathbb {R}^n in terms of powers of the volumes of the individual bodies K K and M M . The special case where K K is the unit ball yields the isoperimetric inequality. In the plane, Betke and Weil (1991) found a sharp upper bound for the mixed area of K K and M M in terms of the perimeters of K K and M M . We extend this result to general dimensions by proving a sharp upper bound for the mixed volume V ( K , M [ n − 1 ] ) V(K,M[n-1]) in terms of the mean width of K K and the surface area of M M . The equality case is completely characterized. In addition, we establish a stability improvement of this and related geometric inequalities of isoperimetric-type.

We prove that no Brunn–Minkowski inequality from the Riemannian theories of curvature-dimension and optimal transportation can be satisfied by a strictly sub-Riemannian structure. Our proof relies on the same method as for the Heisenberg group together with new investigations by Agrachev, Barilari and Rizzi on ample normal geodesics of sub-Riemannian structures and the geodesic dimension attached to them.

Abstract In the paper, we give new improvements of the reverse Hölder and Minkowski integral inequalities. These new results in special case yield the Pólya-Szegö’s inequality and reverse Minkowski’s inequality, respectively.

On the uniqueness and continuity of the dual area measure