Maximizing expected powers of the angle between pairs of points in projective space

Abstract

Among probability measures on $d$-dimensional real projective space, one which maximizes the expected angle $\arccos(\frac{x}{|x|}\cdot \frac{y}{|y|})$ between independently drawn projective points $x$ and $y$ was conjectured to equidistribute its mass over the standard Euclidean basis $\{e_0,e_1,\ldots, e_d\}$ by Fejes Tóth \cite{FT59}. If true, this conjecture evidently implies the same measure maximizes the expectation of $\arccos^α(\frac{x}{|x|}\cdot \frac{y}{|y|})$ for any exponent $α> 1$. The kernel $\arccos^α(\frac{x}{|x|}\cdot \frac{y}{|y|})$ represents the objective of an infinite-dimensional quadratic program. We verify discrete and continuous versions of this {milder} conjecture in a non-empty range $α> α_{Δ^d} \ge 1$, and establish uniqueness of the resulting maximizer $\hat μ$ up to rotation. We show $\hat μ$ no longer maximizes when $α<α_{Δ^d}$. At the endpoint $α=α_{Δ^d}$ of this range, we show another maximizer $μ$ must also exist which is not a rotation of $\hat μ$. For the continuous version of the conjecture, an appendix provided by Bilyk et al in response to an earlier draft of this work combines with the present improvements to yield $α_{Δ^d}<2$. The original conjecture $\ald=1$ remains open (unless $d=1$). However, in the maximum possible range $α>1$, we show $\hat μ$ and its rotations maximize the aforementioned expectation uniquely on a sufficiently small ball in the $L^\infty$-Kantorovich-Rubinstein-Wasserstein metric $d_\infty$ from optimal transportation; the same is true for any measure $μ$ which is mutually absolutely continuous with respect to $\hat μ$, but the size of the ball depends on {$α,d$, and} $\|\frac{d \hat μ}{dμ}\|_{\infty}$.