Related papers: Minkowski dimension for measures
In earlier papers we changed the concept of the inner product to a more general one, to the so-called Minkowski product. This product changes on the tangent space hence we could investigate a more general structure than a Riemannian…
We use a characterization of Minkowski measurability to study the asymptotics of best packing on cut-out subsets of the real line with Minkowski dimension $d\in(0,1)$. Our main result is a proof that Minkowski measurability is a sufficient…
\emph{Minkowski rings} are certain rings of simple functions on the Euclidean space $W = {\mathbb{R}}^d$ with multiplicative structure derived from Minkowski addition of convex polytopes. When the ring is (finitely) generated by a set…
In analogy with the classical Minkowski problem, necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone-volume measure of the unit ball of a finite dimensional Banach space.
An algebraic characterization of vacuum states in Minkowski space is given which relies on recently proposed conditions of geometric modular action and modular stability for algebras of observables associated with wedge-shaped regions. In…
In this article we extend the notion of metric measure spaces to so-called metric two-level measure spaces (m2m spaces): An m2m space $(X, r, \nu)$ is a Polish metric space $(X, r)$ equipped with a two-level measure $\nu \in…
The first author introduced a measure of compactness for families of sets, relative to a class of filters, in the context of convergence approach spaces. We characterize a variety of maps (types of quotient maps, closed maps, and variants…
We establish a Minkowski measurability criterion for a large class of relative fractal drums (or, in short, RFDs), in Euclidean spaces of arbitrary dimension in terms of their complex dimensions, which are defined as the poles of their…
In this article, we present what we believe to be a simple way to motivate the use of Hilbert spaces in quantum mechanics. To achieve this, we study the way the notion of dimension can, at a very primitive level, be defined as the…
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…
A complete family of statistical descriptors for the morphology of large--scale structure based on Minkowski--Functionals is presented. These robust and significant measures can be used to characterize the local and global morphology of…
We give conditions characterizing equality in the Minkowski inequality for big divisors on a projective variety. Our results draw on the extensive history of research on Minkowski inequalities in algebraic geometry.
We propose a measure of shape which is appropriate for the study of a complicated geometric structure, defined using the topology of neighborhoods of the structure. One aspect of this measure gives a new notion of fractal dimension. We…
The notion of the magnitude of a compact metric space was considered in arXiv:0908.1582 with Tom Leinster, where the magnitude was calculated for line segments, circles and Cantor sets. In this paper more evidence is presented for a…
This paper presents a general procedure based on using the method of types to calculate the box dimension of sets. The approach unifies and simplifies multiple box counting arguments. In particular, we use it to generalize the formula for…
A dichotomy for expansions of the real field is established: Either the set of integers is definable or every nonempty bounded nowhere dense definable subset of the real numbers has Minkowski dimension zero.
Magnitude is a numerical isometric invariant of metric spaces, whose definition arises from a precise analogy between categories and metric spaces. Despite this exotic provenance, magnitude turns out to encode many invariants from integral…
We generalize the box and observable distances to those between metric measure spaces with group actions, and prove some fundamental properties. As an application, we obtain an example of a sequence of lens spaces with unbounded dimension…
In this article, the author proposes another way to define the completion of a metric space, which is different from the classical one via the dense property, and prove the equivalence between two definitions. This definition is based on…
Let $C$ be a closed convex cone in ${\mathbb R}^n$, pointed and with interior points. We consider sets of the form $A=C\setminus A^\bullet$, where $A^\bullet\subset C$ is a closed convex set. If $A$ has finite volume (Lebesgue measure),…