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).