Boundedness of pretangent spaces to general metric spaces

TL;DR

This paper establishes a characterization of bounded pretangent spaces in metric spaces through the concept of w-strong porosity of the distance set at zero, linking geometric properties to porosity conditions.

Contribution

It introduces the notion of w-strong porosity at zero and proves its equivalence to the boundedness of all pretangent spaces at a point in a metric space.

Findings

Distance set is w-strongly porous at 0 iff all pretangent spaces are bounded

Provides a new criterion for boundedness of pretangent spaces

Connects porosity properties with geometric structure of metric spaces

Abstract

Let (X,d,p) be a metric space with a metric d and a marked point p. We define the set of w-strongly porous at 0 subsets of [0,\infty) and prove that the distance set {d(x,p): x\in X} is w-strongly porous at 0 if and only if every pretangent space to X at p is bounded.