Mapping radii of metric spaces

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.