Related papers: Minkowski dimension for measures

This paper describes the theory of Minkowski problems for geometric measures in convex geometric analysis. The theory goes back to Minkowski and Aleksandrov and has been developed extensively in recent years. The paper surveys classical and…

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…

It is easy to show that the lower and the upper box dimensions of a bounded set in Euclidean space are invariant with respect to the ambient space. In this article we show that the Minkowski content of a Minkowski measurable set is also…

We discuss the notions of circumradius, inradius, diameter, and minimum width 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…

This work introduces two new notions of dimension, namely the unimodular Minkowski and Hausdorff dimensions, which are inspired from the classical analogous notions. These dimensions are defined for unimodular discrete spaces, introduced in…

We define two notions of discrete dimension based on the Minkowski and Hausdorff dimensions in the continuous setting. After proving some basic results illustrating these definitions, we apply this machinery to the study of connections…

Minkowski space serves as a framework for the theoretical constructions that deal with manifestations of relativistic effects in physical phenomena. But neither Minkowski himself nor the subsequent developers of the relativity theory have…

Magnitude is a numerical invariant of metric spaces introduced by Leinster, motivated by considerations from category theory. This paper extends the original definition for finite spaces to compact spaces, in an equivalent but more natural…

The aim of the paper is to develop a unified algebraical approach to representing the Minkowski difference for convex polyhedra. Namely, there is proposed an exact analytical formulas of the Minkowski difference for convex polyhedra with…

Pairs of metrics in a two-dimensional linear vector space are considered, one of which is a Minkowski type metric. Their simultaneous diagonalizability is studied and canonical presentations for them are suggested.

This article is an introductory work to a larger research project devoted to pure, applied and philosophical aspects of dimension theory. It concerns a novel approach toward an alternate dimension theory foundation: the point-dimension…

In this paper we parallelly build up the theories of normed linear spaces and of linear spaces with indefinite metric, called also Minkowski spaces for finite dimensions in the literature. In the first part of this paper we collect the…

We discuss the definition and measurability questions of random fractals and find under certain conditions a formula for upper and lower Minkowski dimensions. For the case of a random self-similar set we obtain the packing dimension.

Metric mean dimension is a metric invariant of dynamical systems. It is a dynamical analogue of Minkowski dimension of metric spaces. We explain that old ideas of Bowen (1972) can be used for clarifying the local nature of metric mean…

For a collection of convex bodies $P_1,\dots,P_n \subset \mathbb{R}^d$ containing the origin, a Minkowski complex is given by those subsets whose Minkowski sum does not contain a fixed basepoint. Every simplicial complex can be realized as…

We define the notion of weak Minkowski metric and prove some basic properties of such metrics. We also highlight some of the important analogies between Minkowski geometry and the Funk and Hilbert geometries.

We combine functional analytic and geometric viewpoints on approximate Birkhoff and isosceles orthogonality in generalized Minkowski spaces which are finite-dimensional vector spaces equipped with a gauge. This is the first approach to…

This is a detailed introductory survey of the cohomological dimension theory of compact metric spaces.

This article describes the theoretical foundation of and explicit algorithms for a novel approach to morphology and anisotropy analysis of complex spatial structure using tensor-valued Minkowski functionals, the so-called Minkowski tensors.…

We compute the typical (in the sense of Baire's category theorem) multifractal box dimensions of measures on a compact subset of $\mathbb R^d$. Our results are new even in the context of box dimensions of measures.