相关论文: Some elementary aspects of Hausdorff measure and d…

These notes deal with metric spaces, Hausdorff measures and dimensions, Lipschitz mappings, and related topics. The reader is assumed to have some familiarity with basic analysis, which is also reviewed.

We review the motivation and fundamental properties of the Hausdorff dimension of metric spaces and illustrate this with a number of examples, some of which are expected and well-known. We also give examples where the Hausdorff dimension…

In this short note, we show that, in any given metric space, every Lipschitz open-map image of every subset of a given metric space whose boundary is Hausdorff-null is Hausdorff-measurable with respect to the same dimension. The main…

These informal notes briefly discuss some basic topics involving Lipschitz functions, connectedness, and Hausdorff content in particular.

We are going to introduce a new algebraic, analytic structure that is a kind of generalization of the Hausdorff dimension and measure. We give many examples and study the basic properties and relations of such systems.

A number of topics involving metrics and measures are discussed, including some of the special structure associated with ultrametrics.

A Wasserstein spaces is a metric space of sufficiently concentrated probability measures over a general metric space. The main goal of this paper is to estimate the largeness of Wasserstein spaces, in a sense to be precised. In a first…

This document offers a concise introduction to the mathematical theory and practical application of the Hausdorff Measure and Dimension. The primary objective is to clarify and rigorously detail the two most common methods used for…

Two objects may be close in the Hausdorff metric, yet have very different geometric and topological properties. We examine other methods of comparing digital images such that objects close in each of these measures have some similar…

These are some basic notes concerning Holder and Lipschitz classes on metric spaces.

In this paper we examine two basic topological properties of partial metric spaces, namely compactness and completeness. Our main result claims that in these spaces compactness is equivalent to sequential compactness. We also show that…

The standard arithmetic measures of center, the mean and median, have natural topological counterparts which have been widely used in continuum theory. In the context of metric spaces it is natural to consider the Lipschitz continuous…

In this paper we present some bounds of Hausdorff measures of objects definable in o-minimal structures: sets, fibers of maps, inverse images of curves of maps, etc. Moreover, we also give some explicit bounds for semi-algebraic or…

In this paper we show that, given a planar Reifenberg flat domain with small constant and a divergence form operator associated to a real (not necessarily symmetric) uniformly elliptic matrix with Lipschitz coefficients, the Hausdorff…

Given a compact basic semi-algebraic set we provide a numerical scheme to approximate as closely as desired, any finite number of moments of the Hausdorff measure on the boundary of this set. This also allows one to approximate interesting…

This is a survey on recent developments on the Hausdorff dimension of projections and intersections for general subsets of Euclidean spaces, with an emphasis on estimates of the Hausdorff dimension of exceptional sets and on restricted…

Metric spaces are a fundamental component of mathematics and have a paramount importance as a framework for measuring distance. They can be found in many different branches of mathematics, such as analysis and topology. This paper offers an…

We study various measure theories using the classical approach and then compute the Hausdorff dimension of some simple objects and self-similar fractals. We then develop a nonstandard approach to these measure theories and examine the…

This paper presents a comprehensive introduction to the Hausdorff measure, a fundamental tool in fractal geometry and geometric measure theory. We begin by defining the Hausdorff outer measure on subsets of metric spaces, followed by a…

Here we look at (collections of) semimetrics and seminorms, including their ultrametric versions. In particular, we are concerned with geometric properties related to connectedness and topological dimension 0.