Mathematics – Metric Geometry
Scientific paper
2007-04-02
Mathematics
Metric Geometry
24 pages. To appear, Pacific J. Math. Any updates, errata, related references etc. learned of after publication will be noted
Scientific paper
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.
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