If a distance metric is induced by a norm, then centroid can also be used as a measure of central value. Although it is sometimes confused in the applied literature, a distance metric without an inducing norm has no centroid - only the mean, median, and central feature. Note that the mean is independent of distance metric while the central feature is not. Also note that some distance metrics (e.g. Mahalanobis) often produce a centroid equivalent to the mean - which is desirable in some applications and a problem in others.
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I have updated the notebook based on the feedback you have given me, adding a section on different distance functions and calculation methods for the centroid and spatial median. Thank you for your feedback!
Small note: If you weight the distribution centers by metro area populations, the result might change.
Well done, again!
Interesting. I will look into how to do that.