# Mapping radii of metric spaces

## TL;DR

This paper investigates the concept of mapping radii in metric spaces, establishing bounds, developing estimation tools, and calculating specific examples, thereby advancing understanding of geometric embeddings.

## Contribution

It introduces methods for estimating the mapping radius of metric spaces and computes several explicit examples, extending prior geometric bounds.

## Key findings

- The supremum of mapping radii in convex subsets equals the infimum of certain convex combinations.
- Explicit mapping radii are calculated for specific metric spaces.
- Open questions regarding mapping radii are identified.

## 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.