Inequalities and Positive-Definite Functions Arising from a Problem in Multidimensional Scaling

Abstract

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) $n p$ 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.