Chapter 1 Metrics for more than two points at once David H . Wolpert

The conventional definition of a topological metric over a space specifies properties of any measure of “how separated” two points in that space are. Here it is shown how to extend that definition, and in particular the triangle inequality, to concern arbitrary numbers of points. Such a measure of how separated the points within a collection are can be bootstrapped, to measure “how separated” from each other are two (or more) collections. The measure presented here also allows fractional membership of an element in a collection, and therefore measures “how spread out” a probability distribution over a space is. When such a measure is bootstrapped to compare two collections, it measures how separated two probability distributions are, or more generally, how separated a distribution of distributions is.