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

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 ${\mathbf{\Omega }}_{\mathbf{p}}^{\mathbf{X}}(\mathbf{n})$ be the set of pretangent spaces to $X$ at $p$ with normal scaling sequences. We prove that ${\mathbf{\Omega }}_{\mathbf{p}}^{\mathbf{X}}(\mathbf{n})$ 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 ${\mathbf{\Omega }}_{\mathbf{p}}^{\mathbf{X}}(\mathbf{n})$ 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.

24 pages. arXiv admin note: substantial text overlap with arXiv:1302.4599

References in corpus (1)

• Tangent spaces to metric spaces and to their subspaces

Cited by in corpus (1)

• Local one-side porosity and pretangent spaces