Related papers: Average methods and their applications in Differen…
The Minkowski tensors are the natural tensor-valued generalizations of the intrinsic volumes of convex bodies. We prove two complete sets of integral geometric formulae, so called kinematic and Crofton formulae, for these Minkowski tensors.…
Using harmonic mean curvature flow, we establish a sharp Minkowski type lower bound for total mean curvature of convex surfaces with a given area in Cartan-Hadamard 3-manifolds. This inequality also improves the known estimates for total…
The aim of this paper is to study properties of sections of convex bodies with respect to different types of measures. We present a formula connecting the Minkowski functional of a convex symmetric body K with the measure of its sections.…
We study surfaces with parallel normalized mean curvature vector field in Euclidean or Minkowski 4-space. On any such surface we introduce special isothermal parameters (canonical parameters) and describe these surfaces in terms of three…
In this paper we study curvature types of immersed surfaces in three-dimensional (normed or) Minkowski spaces. By endowing the surface with a normal vector field, which is a transversal vector field given by the ambient Birkhoff…
Let $S$ be a compact, orientable surface of hyperbolic type. Let $(k_+,k_-)$ be a pair of negative numbers and let $(g_+, g_-)$ be a pair of marked metrics over $S$ of constant curvature equal to $k_+$ and $k_-$ respectively. Using a…
The classical Minkowski problem in Minkowski space asks, for a positive function $\phi$ on $\mathbb{H}^d$, for a convex set $K$ in Minkowski space with $C^2$ space-like boundary $S$, such that $\phi(\eta)^{-1}$ is the Gauss--Kronecker…
This paper gives a survey of methods for the construction of space-frequency concentrated frames on Riemannian manifolds with bounded curvature, and the applications of these frames to the analysis of function spaces. In this general…
We study a family of inequalities on pairs of measure spaces involving functions defined on product domains. Our main result establishes a Jensen-type inequality under a general product-measure framework, extending classical inequalities…
The Minkowski question mark function is a rich object which can be explored from the perspective of dynamical systems, complex dynamics, metric number theory, multifractal analysis, transfer operators, integral transforms, and as a function…
We introduce a notion of the Euclidean- and the Minkowski rank for arbitrary metric spaces and we study their behaviour with respect to products. We show that the Minkowski rank is additive with respect to metric products, while additivity…
We determine all Finsler metrics of Randers type for which the Riemannian part is a scalar multiple of the Euclidean metric, on an open subset of the Euclidean plane, whose geodesics are circles. We show that the Riemannian part must be of…
In this paper, we first establish an equivalence theorem of Minkowski spaces by using results in centro-affine differential geometry. As an application in Finsler geometry, we gives some new characterizations of Berwald spaces.
We investigate elementary properties of successive radii in generalized Minkowski spaces (that is, with respect to gauges), i.e., we measure the "size" of a given convex set in a finite-dimensional real vector space with respect to another…
The Minkowski functionals are useful statistics to quantify the morphology of various random fields. They have been applied to numerous analyses of geometrical patterns, including various types of cosmic fields, morphological image…
We review some basic concepts related to convex real projective structures from the differential geometry point of view. We start by recalling a Riemannian metric which originates in the study of affine spheres using the Blaschke connection…
Given an Euclidean space, this paper elucidates the topological link between the partial derivatives of the Minkowski functional associated to a set (assumed to be compact, convex, with a differentiable boundary and a non-empty interior)…
We propose a novel method for the description of spatial patterns formed by a coverage of point sets representing galaxy samples. This method is based on a complete family of morphological measures known as Minkowski functionals, which…
Our previous exploration of the $\cE_g^{PD}$-geometry has shown that the field is promising. Namely, the $\cE_g^{PD}$-approach is amenable to development of novel trends in relativistic and metric differential geometry and can particularly…
We consider the family of convex bodies obtained from an origin symmetric convex body $K$ by multiplication with diagonal matrices, by forming Minkowski sums of the transformed sets, and by taking limits in the Hausdorff metric. Support…