Mapping radii of metric spaces

Abstract

It is known that every closed curve of length \leq 4 in R^n (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 R^n 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\in X). Several explicit mapping radii are calculated, and open questions noted.

24 pages. To appear, Pacific J. Math. Any updates, errata, related references etc. learned of after publication will be noted at http://math.berkeley.edu/~gbergman/papers . Changes from first version: correction to display (30) and statement of Cor. 26; new display (41) and Lemma 33; minor clarifications of wording etc.