Spheres in Infinite-Dimensional Normed Spaces are Lipschitz Contractible

On the Geometry of Spheres in Normed Linear Spaces

Some simplifications of Schaffer's girth and perimeter of the unit spheres are introduced. Their general properties are discussed, and they are used to study the lp, Lp spaces, uniformly nonsquare spaces, and their isomorphic classes. 1980 Mathematics subject classification (Amer. Math. Soc.) (1985 Revision): 46 B 20.

Note on bi-Lipschitz embeddings into normed spaces

Let (X, d), (Y, ρ) be metric spaces and f : X → Y an injective mapping. We put ‖f‖Lip = sup{ρ(f(x), f(y))/d(x, y); x, y ∈ X, x 6= y}, and dist(f) = ‖f‖Lip.‖f ‖Lip (the distortion of the mapping f). We investigate the minimum dimension N such that every n-point metric space can be embedded into the space lN ∞ with a prescribed distortion D. We obtain that this is possible for N ≥ C(logn)2n3/D, w...

Euclidean Structure in Finite Dimensional Normed Spaces

Finite Connected H-spaces Are Contractible

The non-Hausdorff suspension of the one-sphere S1 of complex numbers fails to model the group’s continuous multiplication. Moreover, finite connected H-spaces are contractible, and therefore cannot model infinite connected non-contractible H-spaces. For an H-space and a finite model of the topology, the multiplication can be realized on the finite model after barycentric subdivision.

Differentiability Properties of -stable Vector Functions in Infinite-dimensional Normed Spaces

The aim of this paper is to continue the study of properties of an -stable at a point vector function. We show that any -stable at a point function from arbitrary normed linear space is strictly differentiable at the considered point.