Summary: We solve the following variational problem: Find the maximum of \(E|X-Y |\) subject to \(E|X|^2 \leq 1\), where \(X\) and \(Y\) are i.i.d. random \(n\)-vectors, and \(|\cdot |\) is the usual Euclidean norm on \(\mathbb{R}^n\). This problem arose from an investigation into multidimensional scaling, a data analytic method for visualizing proximity data. We show that the optimal \(X\) is unique and is (1) uniform on the surface of the unit sphere, for dimensions \(n\geq 3\), (2) circularly symmetric with a scaled version of the radial density \(\rho/ (1- \rho^2 )^{1/2}\), \(0\leq \rho\leq 1\), for \(n=2\), and (3) uniform on an interval centered at the origin, for \(n=1\) (Plackett’s theorem). By proving spherical symmetry of the solution, a reduction to a radial problem is achieved. The solution is then found using the Wiener-Hopf technique for (real) \(n<3\). The results are reminiscent of classical potential theory, but they cannot be reduced to it.
Along the way, we obtain results of independent interest: for any i.i.d. random \(n\)-vectors \(X\) and \(Y\), \[ E|X-Y |\leq E|X+Y |. \] Further, the kernel \[ K_{p, \beta} (x,y)= |x+y |_p^\beta- |x-y |_p^\beta, \quad x,y\in \mathbb{R}^n \qquad \text{ and} \quad |x|_p= \bigl( \sum|x_i |^p \bigr)^{ 1/p}, \] is positive-definite, that is, it is the covariance of a random field, \(K_{p, \beta} (x,y)= E[Z(x) Z(y) ]\) for some real- valued random process \(Z(x)\), for \(1\leq p\leq 2\) and \(0< \beta\leq p\leq 2\) (but not for \(\beta> p\) or \(p> 2\) in general). Although this is an easy consequence of known results, it appears to be new in a strict sense.
In the radial problem, the average distance \(D(r_1, r_2)\) between two spheres of radii \(r_1\) and \(r_2\) is used as a kernel. We derive properties of \(D(r_1, r_2)\), including nonnegative definiteness on signed measures of zero integral.