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.
Cite
@article{arxiv.0704.0275,
title = {Mapping radii of metric spaces},
author = {George M. Bergman},
journal= {arXiv preprint arXiv:0704.0275},
year = {2021}
}
Comments
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.