Monotonicity of the mean distance for empirical dependent Gaussian samples (Q1124246)

Let X, Y be i.i.d. symmetric Gaussian random vectors with values in \(R^ n\) and covariance matrix K. Denote \(\phi (x,y)=\min_{\sigma}\| x- \sigma y\|_ 1\), where \(x,y\in R^ n\), \(\| \cdot \|_ 1\) is the \(\ell_ 1\)-norm and \(\sigma\) runs over the group of all permutations. The following theorem is investigated. If K is bounded from above by the identity matrix I then the expectation E \(\phi\) (X,Y) attains its maximum for \(K=I.\) This theorem was announced in the author's paper, ibid. 142, 164-166 (1986; see the preceding review, Zbl 0678.62055), but there was a gap in the proof. Here, a corrected version of the proof is sketched.