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The earth mover's distance (EMD) is a well-known metric on spaces ofhistograms; roughly speaking, the EMD measures the minimum amount of workrequired to equalize two histograms. The EMD has a natural generalization thatcompares an arbitrary number of histograms; in this case, the EMD can be viewedas hypervolume in $d$ dimensions, where the histograms are vertices of a$d$-simplex. For $d=2$, it is known that the EMD between three histogramsequals half the sum of the pairwise EMDs -- a sort of Heron's formula forhistograms, but where the area equals the semiperimeter. In this paper, byintroducing an object we call the earth mover's simplex, we prove twogeneralizations of this Heron-like formula in arbitrary dimension: the first (asort of Cayley-Menger formula) expresses the EMD in terms of the edge lengths(the pairwise EMDs), the second in terms of the facets (EMDs excluding onehistogram).

A Cayley-Menger formula for the earth mover's simplex