Mapping radii of metric spaces

George M. Bergman

Vol. 236 (2008), No. 2, 223–261

DOI: 10.2140/pjm.2008.236.223

Abstract

It is known that every closed curve of length ≤ 4 in ℝⁿ for n > 0 can be surrounded by a sphere of radius 1, and that this is the best bound. Letting S denote the circle of circumference 4, with the arc-length metric, we here express this fact by saying that the mapping radius of S in ℝⁿ is 1.

Tools are developed for estimating the mapping radius of a metric space X in a metric space Y . In particular, it is shown that for X a bounded metric space, the supremum of the mapping radii of X in all convex subsets of normed metric spaces is equal to the infimum of the sup norms of all convex linear combinations of the functions d(x,⋅) : X → ℝ (x ∈ X).

Several explicit mapping radii are calculated, and open questions noted.

Keywords

nonexpansive map between metric spaces, maximum radius of image, convex subset of a normed vector space

Mathematical Subject Classification 2000

Primary: 54E40

Secondary: 46B20, 46E15, 52A40

Milestones

Received: 8 April 2007

Revised: 5 July 2007

Accepted: 24 March 2008

Published: 1 June 2008