Mapping Radii of Metric Spaces
نویسندگان
چکیده
It is known that every closed curve of length ≤ 4 in Rn (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. 1 The definition, and three examples. Definition 1. We will denote by Metr the category whose objects are metric spaces, and whose morphisms are nonexpansive maps. That is, for metric spaces X and Y we let (1) Metr(X,Y ) = {f : X → Y | (∀x0, x1 ∈ X) d(f(x0), f(x1)) ≤ d(x0, x1)}. Throughout this note, a map of metric spaces will mean a morphism in Metr. Given a nonempty subset A of a metric space Y, we define its radius by (2) radY (A) = infy∈Y supa∈A d(a, y), a nonnegative real number or +∞. For metric spaces X and Y, we define the mapping radius of X in Y by (3) map-rad(X,Y ) = supf∈Metr(X,Y ) radY (f(X)) = supf∈Metr(X,Y ) infy∈Y supx∈X d(f(x), y). If X is a metric space and Y a class of metric spaces, we likewise define (4) map-rad(X,Y) = supY ∈Y map-rad(X,Y ) = supY ∈Y, f∈Metr(X,Y ) infy∈Y supx∈X d(f(x), y). (The term “mapping radius” occurs occasionally in complex analysis with an unrelated meaning [15, Def. 7.11].) All vector spaces in this note will be over the field of real numbers unless the contrary is stated. The result stated in the first sentence of the abstract has been discovered many times [5], [6], [18], [19], [25]. (Usually, the length of the closed curve is given as 1 and the radius of the sphere as 1/4, but the scaled-up version will be more convenient here.) Let us obtain it in somewhat greater generality. ∗2000 Mathematics Subject Classifications. Primary: 54E40. Secondary: 46B20, 46E15, 52A40.
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