Minkowski Inequality Research Articles

New weighted Alexandrov-Fenchel type inequalities and Minkowski inequalities in space forms

Abstract The classical Minkowski inequality implies that the volume of a bounded convex domain in the Euclidean space is controlled from above by the integral of the mean curvature of its boundary. In this note, an analogous inequality is established without assuming convexity, valid for all bounded smooth domains in a complete manifold whose bottom spectrum is suitably large relative to its Ricci curvature lower bound. An immediate consequence is the nonexistence of embedded closed minimal hypersurfaces in such manifolds. The same nonexistence issue is also addressed for steady and expanding Ricci solitons. The proofs are very much inspired by a sharp monotonicity formula, derived for positive harmonic functions on manifolds with positive spectrum.

Minkowski inequality in GRW spacetimes

Formulation of the Problem. A large amount of mathematical literature is devoted to classical inequalities. Helder's inequalities, a special case of which is the Cauchy-Buniakovsky inequality, as well as Minkowski's, which is a polygon inequality in a normed space, underlie the geometry of unitary and normed spaces - finite and infinite-dimensional (Banach). The article considers the generalization of these constructions - both in discrete form, that is, for finite sums and series, and for integrals. It is essential that inequalities for sums are proved by elementary methods, without the use of differential calculus. The results obtained can be used in scientific activities for evaluating some expressions in the form of sums or integrals, as well as by students in preparation for Olympiads and even for studying mathematics in school circles. Materials and Methods. To prove the generalized Minkowski inequality and the integral inequalities of Helder and Minkowski, the generalized Helder inequality for sums, which was previously obtained by the author which, in turn, was derived from Cauchy's inequality. Results. The generalized Minkowski inequalities were proved for finite sums and infinite series with non-negative members and the integral for non-negative functions, as well as the generalized integral Helder inequality and, in a special case, the Cauchy-Bunyakovsky inequality. Conclusion. The application of the generalized Helder and Minkowski inequalities for sums, series, and integrals is a fairly effective method that allows you to obtain interesting consequences, important estimates – you only need to successfully select finite-dimensional or infinite-dimensional vectors or functions and apply the proved inequalities to them. On this path, there is a great deal of space for creative activity.

Discrete Brunn–Minkowski inequality for subsets of the cube

Abstract This article describes two anisotropic area-preserving flows for plane curves, both of which are considered to deform one convex curve into another. Different monotonic entropy functions are identified under these flows, which can be utilized to derive two significant entropy inequalities: the log-Minkowski inequality and the curvature entropy inequality, as well as the Brunn–Minkowski inequality.

Abstract The infinitesimal forms of the $$L_{p}$$ L p -Brunn–Minkowski inequalities for variational functionals, such as the q-capacity, the torsional rigidity, and the first eigenvalue of the Laplace operator, are investigated for $$p \ge 0$$ p ≥ 0 . These formulations yield Poincaré-type inequalities related to these functionals. As an application, the $$L_{p}$$ L p -Brunn–Minkowski inequalities for torsional rigidity with $$0 \le p < 1$$ 0 ≤ p < 1 are confirmed for small smooth perturbations of the unit ball.

In the paper, we establish a volumetric Minkowski inequality for complete manifolds admitting a weighted Poincaré inequality with the weight commensurable to the Ricci curvature lower bound. More precisely, we show that the weighted volume of a compact smooth domain in such manifolds is bounded from above by an integral of the mean curvature of its boundary. In particular, this implies that such manifolds admit no compact embedded minimal hypersurfaces. Examples of such manifolds abound and include both the manifolds with positive bottom spectrum and a large family of complete manifolds with nonnegative Ricci curvature, thus unifying the results by Munteanu and Wang in Munteanu, O., Wang, J.: A Minkowski type inequality for manifolds with positive spectrum. arXiv:2309.13749 (2023) and by Benatti, Fogagnolo and Mazzieri in Benatti, L., Fogagnolo, M., Mazzieri, L.: Minkowski inequality on complete Riemannian manifolds with nonnegative Ricci curvature. arXiv:2101.06063 (2021).

In this paper, our first objective is to define the idea of grand variable Herz spaces. Then, our main goal is to prove boundedness results for operators, including the rough Riesz potential operator of variable order and the fractional Hardy operators, on grand variable Herz spaces under some proper assumptions. To prove the boundedness results, we use Holder-type and Minkowski inequalities. In the proof of the main result, we use different techniques. We divide the summation into different terms and estimate each term under different conditions. Then, by combining the estimates, we prove that the rough Riesz potential operator of variable order and the fractional Hardy operators are bounded on grand variable Herz spaces. It is easy to show that the rough Riesz potential operator of variable order generalizes the Riesz potential operator and that the fractional Hardy operators are generalized versions of simple Hardy operators. So, our results extend some previous results to the more generalized setting of grand variable Herz spaces.

We present a new proof of the Willmore inequality for an arbitrary bounded domain Ω ⊂ R n \Omega \subset \mathbb {R}^{n} with smooth boundary. Our proof is based on a parametric geometric inequality involving the electrostatic potential for the domain Ω \Omega ; this geometric inequality is derived from a geometric differential inequality in divergence form. Our parametric geometric inequality also allows us to give new proofs of the quantitative Willmore-type and the weighted Minkowski inequalities by Agostiniani and Mazzieri.

Abstract In this article, we extend the notion of multiplicity for weakly graded families of ideals which are bounded below linearly. In particular, we show that the limit exists where is a bounded below linearly weakly graded family of ideals in a Noetherian local ring of dimension with . Furthermore, we prove that “volume = multiplicity” formula and Minkowski inequality hold for such families of ideals. We explore some properties of for weakly graded families of ideals of the form where is an ‐primary graded family of ideals. We provide a necessary and sufficient condition for the equality in Minkowski inequality for the weakly graded families of ideals of the form where is a bounded filtration. Moreover, we generalize a result of Rees characterizing the inclusion of ideals with the same multiplicities for the above families of ideals. Finally, we investigate the asymptotic behavior of the length function where is a filtration of ideals (not necessarily ‐primary).

ABSTRACTThe significance of the Jensen inequality stems from its impactful and compelling outcomes. As a generalization of classical convexity, it plays a key role in deriving other well‐known inequalities such as Hermite–Hadamard, Hölder, Minkowski, arithmetic‐geometric, and Young's inequalities. So, this inequality has become an influential concept in a wide range of scientific fields. Besides, interval analysis provides methods for managing uncertainty in data, making it possible to build mathematical and computer models of various deterministic real‐world phenomena. In this paper, taking into account all of these, we first present several refinements of the Jensen inequality for the left and right convex interval‐valued functions. We also provide examples with corresponding graphs to demonstrate these refinements more clearly. Next, we adopt a novel approach to derive several bounds for the Jensen gap in integral form using the gH‐differentiable interval valued functions as well as various related notions. Moreover, we obtain the proposed bounds by utilizing the renowned Ostrowski inequality. The fundamental benefit of the newly discovered inequalities is that they extend to many known inequalities in the literature, as discussed in this work.

Conformable integrals and derivatives have received more attention in recent years as a means of determining different kinds of inequalities. In the research work, we define a novel class of (k, ρ)-conformable fractional integrals ((k, ρ)-CFI). Also, we establish the refinement of the reverse Minkowski inequality incorporating the (k, ρ)-conformable fractional integral operators. The proposed (k, ρ)-conformable fractional integral operators are used to present the two new theorems that correlate with this inequality, along with declarations and verifications of other inequalities. The inequalities presented in this work are more general as compared to the existing literature. The special cases of our main findings are given in the paper.

In 2003, associated with the radial Minkowski additions of star bodies, Zhao and Leng established the dual Brunn–Minkowski inequality for intersection bodies. In this paper, associated with the L_{p} -radial Minkowski combinations of star bodies, we firstly prove the L_{p} -dual Brunn–Minkowski inequality for intersection bodies. Further, associated with the L_{p} -Minkowski combinations of convex bodies, we give the L_{p} -Brunn–Minkowski inequality for star dualities of intersection bodies.

abstract: We reinterpret renormalized volume as the asymptotic difference of the isoperimetric profiles for convex co-compact hyperbolic 3-manifolds. By similar techniques we also prove a sharp Minkowski inequality for horospherically convex sets in $\mathbb{H}^3$. Finally, we include the classification of stable constant mean curvature surfaces in regions bounded by two geodesic planes in $\mathbb{H}^3$ or in cyclic quotients of $\mathbb{H}^3$.

In this note, a generalization by Yuan and Leng of Minkowski's determinant inequality is improved. An interpolation of the Yuan and Leng's inequality is shown by using the negativity of some related functional. Some refined versions of Minkowski's inequality and of Ky Fan's inequality are presented.

In the case of symmetries with respect to 𝑛 independent linear hyperplanes, a stability versions of the Logarithmic Brunn–Minkowski Inequality and the Logarithmic Minkowski Inequality for convex bodies are established.

The Dual Orlicz Brunn–Minkowski Inequality for the Polars of Mixed Projection Bodies

This study presents a novel approach to metric spaces through the lens of geometric calculus, redefining traditional structures with new operations and properties derived from non-Newtonian measures. Specifically, we develop and prove geometric versions of the Hölder and Minkowski inequalities, which provide foundational support for applying these spaces in analysis. Additionally, we establish key relationships between geometric and classical metric spaces, examining concepts such as openness, closedness, and separability within this geometric framework. By exploring topological characteristics and separability conditions in geometric metric spaces, this work enhances the understanding of metric spaces' structural properties, offering potential applications in fields that require flexible metric adaptations, such as data science, physics, and computational geometry. This framework's adaptability makes it relevant for scenarios where non-Euclidean or high-dimensional spaces are needed, allowing for versatile applications and extending classical metric concepts into broader analytical contexts.

Companions to the Brunn–Minkowski inequality