Uniform boundedness of pretangent spaces, local constancy of metric derivatives and strong right upper porosity at a point

TL;DR

This paper establishes a connection between the uniform boundedness of pretangent spaces at a point in a metric space and the porosity of the distance set, linking it to metric derivatives of differentiable mappings.

Contribution

It proves that the uniform boundedness of pretangent spaces is equivalent to the complete strong porosity of the distance set and to the constancy of metric derivatives in the space.

Findings

Uniform boundedness of pretangent spaces characterized by porosity.

Equivalence between bounded pretangent spaces and constant metric derivatives.

Connection between local porosity and differentiability properties.

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 $(r_n), r_n\downarrow 0.$ A scaling sequence $(r_n)$ is normal if this sequence is eventually decreasing and there is $(x_n)$ such that $\mid d(x_n,p)-r_n\mid=o(r_n)$ for $n\to\infty.$ Let $\mathbf{\Omega_{p}^{X}(n)}$ be the set of pretangent spaces to $X$ at $p$ with normal scaling sequences. We prove that $\mathbf{\Omega_{p}^{X}(n)}$ is uniformly bounded if and only if $\{d(x,p): x\in X\}$ is a so-called completely strongly porous set. It is also proved that the uniform boundedness of $\mathbf{\Omega_{p}^{X}(n)}$ is an equivalent of the constancy of metric derivatives of all metrically differentiable mappings on $X$ in the open balls of a fixed radius centered at the marked points of pretangent spaces.