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 (xn), xn ∈ X, whose degree of convergence is comparable with a given scaling sequence (rn), rn 0. A scaling sequence (rn) is normal if this sequence is eventually decreasing and there is (xn) such that d(xn,p)-rn=o(rn) for n∞. Let pX(n) be the set of pretangent spaces to X at p with normal scaling sequences. We prove that pX(n) is uniformly bounded if and only if \d(x,p): x∈ X\ is a so-called completely strongly porous set. It is also proved that the uniform boundedness of pX(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.