A continuous extension operator for convex metrics
by
Ihor Stasyuk
Lviv National University, University of Saskatchewan
Coauthors: E.D. Tymchatyn

The problem of extension of metrics has a long history. Various results on extensions of (pseudo)metrics (which in particular satisfy special properties) were obtained by many authors (see for instance [BB], [B1], [TZ], [TZ1]. We consider the problem of simultaneous extension of continuous convex metrics defined on subcontinua of a Peano continuum.

A metric r on a Peano continuum X is said to be convex if for each x, y ∈ X there is an arc [xy] with endpoints x and y such that [xy] is isometric to the closed interval [0, r(x, y)] in the real line R. It is known [B] and [M] that a metric continuum is locally connected if and only if it has a convex metric.

If X is a Peano continuum and A is a locally connected subcontinuum of X let CM(A) denote the set of continuous convex metrics on A. We identify each metric r ∈ CM(A) with its graph which is a compact subset of the space X×X×R. Let

CM= È {CM(A): A is a Peano subcontinuum of X}.

We assume that the distance between two metrics is the Hausdorff distance between their graphs. It follows from Bing [B] that for every convex metric on a subcontinuum of a Peano continuum there exists a convex extension onto the whole space. We modify Bing's construction in order to obtain a continuous simultaneous extension operator. The following theorem is our main result.

Theorem. Let X be a Peano continuum. There is a continuous extension operator u: CM→ CM(X).

References

[BB] T. Banakh, C. Bessaga, On linear operators extending [pseudo]metrics, Bull. Polish Acad. Sci. Math. 48 (2000), no. 1, 35-49.

[B1] R.H. Bing, A convex metric for a locally connected continuum, Bull. Amer. Math. Soc. 55 (1949), 812-819.

[B] R.H. Bing, Partitioning continuous curves, Bull. Amer. Math. Soc. 58 (1952), 536-556.

[M] E. E. Moise, Grille decomposition and convexification theorems for compact metric locally connected continua, Bull. Amer. Math. Soc. 55 (1949), 1111-1121.

[TZ] E.D. Tymchatyn, M. Zarichnyi, On simultaneous linear extensions of partial (pseudo)metrics, Proc. Amer. Math. Soc. 132 (2004), 2799-2807.

[TZ1] E.D. Tymchatyn, M. Zarichnyi, A note on operators extending partial ultrametrics, Comment. Math. Univ. Carolinae 46 (2005), no. 3, 515-524.

Date received: May 5, 2008