Mathematicians and engineers are disinclined to agree about anything in public: should the area of a circle be described using the neat formula or in terms of the more easily measured diameter as , for example?
— J. Bryant and C. Sangwin, How Round is Your Circle?
I will argue on behalf of the diameter but from a mathematician’s perspective. The diameter of a nonempty set is
Whether should be or I’ll leave for you to decide. The radius of can be defined as
For a circle — whether this word means or — these definitions indeed agree with the diameter and radius. The example of shows that in the definition of the radius we should not require .
The problem of determining the radius of a given set was posed in 1857 by J.J.Sylvester in Quarterly Journal of Pure and Applied Mathematics. Thanks to Google Books, I can reproduce his article in its entirety:
Suppose that is a map of that is nonexpanding/short/metric/1-Lipschitz or whatyoucallit: for all . Clearly, the diameter does not increase: . What happens to the radius is not nearly as obvious…
It turns out that the radius does not increase either. Indeed, by Kirszbraun’s theorem can be extended to a 1-Lipschitz map of the entire plane, and the extended map tells us where the center of a bounding circle should go. Kirszbraun’s theorem is valid in for every , as well as in a Hilbert space. Hence, nonexpanding maps do not increase the radius of any subset of a Hilbert space.
However, general normed vector spaces are different…
The example given below is wrong; the map is not 1-Lipschitz. I keep it for historical record.
For example, consider the taxicab distance in the plane. The 4-point set in red is mapped by isometrically in such a way that .
This example is as bad as it gets: for any subset of a metric space we have
provided that is a nonexpanding map into a metric space .
The extra factor of 2 in causes problems when one wants control some quantity under iteration of . There’s a big difference between and .