for all a, b ∈ E and 0 ≤ t ≤ 1. Equivalently, f is affine if the map T :E → F , defined by Tx = fx− f(0), is linear. An isometry need not be affine. To see this, let E be the real line R, let F be… (More)

The upper set Ã of a metric space A is a subset of A × (0,∞) , consisting of all pairs (x, |x − y|) with x, y ∈ A , x = y . We consider various properties of Ã and a metric of Ã , called the broken… (More)

We generalize the concept of a uniform domain in Banach spaces into two directions. (1) The ordinary metric d of a domain is replaced by a metric e ≥ d, in particular, by the inner metric of the… (More)

A domain G in a Banach space is said to be δ -hyperbolic if it is a Gromov δ -hyperbolic space in the quasihyperbolic metric. Then G has the Gromov boundary ∂G and the norm boundary ∂G . We show that… (More)

A set A in R" is called porous if there is a > 0 such that every ball B(x,r) contains a point whose distance from A is at least ar. We show that porosity is preserved by quasisymmetric maps, in… (More)

We show that quasihyperbolic geodesics exist in convex domains in reflexive Banach spaces and that quasihyperbolic geodesics are quasiconvex in the norm metric in convex domains in all normed spaces.… (More)

A criterion for normality and compactness of families of K -quasiregular mappings of bounded multiplicity is established and then applied to the study of the branch set and its image.

1.1. Notation. Throughout this article, E and F are real Banach spaces (sometimes Hilbert spaces or just euclidean spaces) of dimension at least one. The norm of a vector x is written as |x|. In a… (More)

The paper deals with basic smoothness and bilipschitz properties of geodesics, balls and spheres in the quasihyperbolic metric of a domain in a Hilbert space. 2000 Mathematics Subject Classification:… (More)