Uniform boundedness of pretangent spaces and local strong one-side porosity

TL;DR

This paper investigates the conditions under which pretangent spaces to a metric space at a point are uniformly bounded, linking this property to the local strong porosity of the distance set from that point.

Contribution

It establishes a characterization of uniform boundedness of pretangent spaces in terms of complete strong porosity of the distance set.

Findings

Pretangent spaces are uniformly bounded iff the distance set is completely strongly porous.

Introduces the concept of normal scaling sequences for pretangent spaces.

Connects geometric properties of the space with porosity conditions.

Abstract

Let (X,d,p) be a pointed metric space. A pretangent space to X at p is a metric space consisting of some equivalence classes of convergent to p sequences (x_n), x_n \in X, whose degree of convergence is comparable with a given scaling sequence (r_n), r_n\downarrow 0. We say that (r_n) is normal if there is (x_n) such that |d(x_n,p)-r_n|=o(r_n) for n\to\infty. Let Omega_{p}^{X}(n) be the set of pretangent spaces to X at p with normal scaling sequences. We prove that the spaces from Omega_{p}^{X}(n) are uniformly bounded if and only if {d(x,p:x\in X}is a so-called completely strongly porous set.