On Bilipschitz Extensions in Real Banach Spaces

and Applied Analysis 3 (3) k D (z 1 , z 2 ) ≤ c 󸀠 1 log(1+|z 1 −z 2 |/min{d D (z 1 ), d D (z 2 )})+ d for all z 1 , z 2 ∈ D. Gehring and Palka [14] introduced the quasihyperbolic metric of a domain in R, and it has been recently used by many authors in the study of quasiconformal mappings and related questions [16]. In the case of domains in R, the equivalence of items (1) and (3) in Theorem E is due to Gehring and Osgood [17] and the equivalence of items (2) and (3) is due to Vuorinen [18]. Many of the basic properties of this metric may be found in [4, 5, 17]. Recall that an arc α from z 1 to z 2 is a quasihyperbolic geodesic if l k (α) = k D (z 1 , z 2 ). Each subarc of a quasihyperbolic geodesic is obviously a quasihyperbolic geodesic. It is known that a quasihyperbolic geodesic between every pair of points in E exists if the dimension of E is finite, see [17, Lemma 1]. This is not true in arbitrary spaces (cf. [19, Example 2.9]). In order to remedy this shortage, Väisälä introduced the following concepts [5]. Definition 4. Letα be an arc inE.The arcmay be closed, open, or half open. Let x = (x 0 , . . . , x n ), n ≥ 1, be a finite sequence of successive points of α. For h ≥ 0, we say that x is h-coarse if k D (x j−1 , x j ) ≥ h for all 1 ≤ j ≤ n. LetΦ k (α, h) be the family of all h-coarse sequences of α. Set