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

TL;DR

This paper investigates probability measures on real projective space that maximize the expected angle between points, verifying a conjecture for certain exponents and establishing uniqueness and stability properties of these measures.

Contribution

It verifies a milder version of a conjecture about maximizing expected angles in projective space for a range of exponents and proves uniqueness and local optimality of the maximizers.

Findings

Confirmed the conjecture for ppa > lpha_{\u2206^d}  and established measure uniqueness.

Showed the measure no longer maximizes when lpha < lpha_{\u2206^d}.

Proved local maximality in the  Kantorovich-Rubinstein-Wasserstein metric for lpha > 1.

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 T\'oth \cite{FT59}. If true, this conjecture evidently implies the same measure maximizes the expectation of $\arccos^\alpha(\frac{x}{|x|}\cdot \frac{y}{|y|})$ for any exponent $\alpha > 1$. The kernel $\arccos^\alpha(\frac{x}{|x|}\cdot \frac{y}{|y|})$ represents the objective of an infinite-dimensional quadratic program. We verify discrete and continuous versions of this {milder} conjecture in a non-empty range $\alpha > \alpha_{\Delta^d} \ge 1$, and establish uniqueness of the resulting maximizer $\hat \mu$ up to rotation. We show $\hat \mu$ no longer maximizes when $\alpha<\alpha_{\Delta^d}$. At the endpoint $\alpha=\alpha_{\Delta^d}$ of this range, we show another maximizer $\mu$ must also exist which is not a rotation of $\hat \mu$. 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 $\alpha_{\Delta^d}<2$. The original conjecture $\ald=1$ remains open (unless $d=1$). However, in the maximum possible range $\alpha>1$, we show $\hat \mu$ 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 $\mu$ which is mutually absolutely continuous with respect to $\hat \mu$, but the size of the ball depends on {$\alpha,d$, and} $\|\frac{d \hat \mu}{d\mu}\|_{\infty}$.