Reducing multidimensional two-sample data to one-dimensional interpoint comparisons

The most popular technique for reducing the dimensionality in comparing two multidimensional samples of X ~ F and Y~ G is to analyze distributions of interpoint comparisons based on a univariate function h (e.g. the interpoint distances). We provide a theoretical foundation for this technique, by showing that having both i) the equality of the distributions of within sample comparisons (h(X₁, X₂) =L h(Y₁, Y₂)) and ii) the equality of these with the distribution of between sample comparisons ((h(X₁, X₂) =L h(X₃, Y₃)) is equivalent to the equality of the multivariate distributions $(F = G)$.