Related papers: A Remark on the Anisotropic Outer Minkowski conten…

This paper presents a constructive proof of the existence of a regular non-atomic strictly-positive measure on any second-countable non-atomic locally compact Hausdorff space. This construction involves a sequence of finitely-additive set…

In this paper we define a new class of metric spaces, called multi-model Cantor sets. We compute the Hausdorff dimension and show that the Hausdorff measure of a multi-model Cantor set is finite and non-zero. We then show that a bilipschitz…

The norm of Cartan torsion plays an important role for studying of immersion theory in Finsler geometry. Indeed, Finsler manifold with unbounded Cartan torsion can not be isometrically imbedded into any Minkowski space. In this paper, we…

In this paper we present a surprisingly short proof of Minkowski's second theorem. The author hopes there is no mistake in it, though the argument seems to be too plain to contain one. Also, we apply the main construction of the proof to…

A Besicovitch set is a set which contains a unit line segment in any direction. It is known that the Minkowski and Hausdorff dimensions of such a set must be greater than or equal to 5/2 in \R^3. In this paper we show that the Minkowski…

This paper focuses on extending the use of Minkowski Tensors to analyze anisotropic signals in cosmological data, focusing on those introduced by redshift space distortion. We derive the ensemble average of the two translation-invariant,…

Let R a be countable ergodic equivalence relation of type II_1 on a standard probability space (X,m). The group Out(R) of outer automorphisms of R consists of all invertible Borel measure preserving maps of the space which map R-classes to…

We present a class of exact solutions of Einstein's gravitational field equations describing spherically symmetric and static anisotropic stellar type configurations. The solutions are obtained by assuming a particular form of the…

We prove that an arbitrary convex body $C \subseteq \mathbf{R}^{n+1} $, whose $ k $-th anisotropic curvature measure (for $ k =0, \ldots , n-1 $) is a multiple constant of the anisotropic perimeter of C, must be a rescaled and translated…

The interrelations between (upper and lower) Minkowski contents and (upper and lower) surface area based contents (S-contents) as well as between their associated dimensions have recently been investigated for general sets in R^d (cf. [3]).…

We provide a new class of interior solutions for anisotropic stars admitting conformal motion. The Einstein's field equations in this construction are solved for specific choices of the density/mass functions. We analyze the behavior of the…

We prove the existence and uniqueness up to translations of the solution to a Minkowski type problem for the torsional rigidity in the class of open bounded convex subsets of the $n$-dimensional Euclidean space. For the existence part we…

The $L_p$-Minkowski problem deals with the existence of closed convex hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed $p$-area measures. It extends the classical Minkowski problem and embraces several important geometric and physical…

In this work we study an anisotropic model of general relativity based on the framework of Finsler geometry. The observed anisotropy of the microwave background radiation is incorporated in the Finslerian structure of space-time. We also…

The Brunn-Minkowski inequality states that for bounded measurable sets $A$ and $B$ in $\mathbb{R}^n$, we have $|A+B|^{1/n} \geq |A|^{1/n}+|B|^{1/n}$. Also, equality holds if and only if $A$ and $B$ are convex and homothetic sets in…

After giving a general introduction to the main known results on the anisotropic Calder{\'o}n problem on n-dimensional compact Riemannian manifolds with boundary, we give a motivated review of some recent non-uniqueness results obtained in…

We have proposed a generally covariant non-relativistic particle model that can represent the $\kappa$-Minkowski noncommutative spacetime. The idea is similar in spirit to the noncommutative particle coordinates in the lowest Landau level.…

We describe the harmonic interpolation of convex bodies, and prove a strong form of the Brunn-Minkowski inequality and characterize its equality case. As an application we improve a theorem of Berndtsson on the volume of slices of a…

We propose to derive deviation measures through the Minkowski gauge of a given set of acceptable positions. We show that, given a suitable acceptance set, any positive homogeneous deviation measure can be accommodated in our framework. In…

This manuscript explores the compact geometries by employing Karmarkar condition with the charged anisotropic source of matter distribution. For this purpose, we consider an explicit model by indulging $\mathrm{g}_{rr}$ metric potential…