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

26 Jul 2020  ·  Tongseok Lim, Robert J. McCann ·

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 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 conjecture in a non-empty range $\alpha > \alpha_{\Delta^d} \geq 1$, and establish uniqueness of the resulting maximizer $\hat \mu$ up to rotation. 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 the present work allows us to infer $\alpha_{\Delta^d} <2$. The original conjecture $\alpha_{\Delta^d}=1$ remains open (unless $d=1$). Moreover, in the full 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 $\|\frac{d \hat \mu}{d\mu}\|_{\infty}$. read more

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Metric Geometry Mathematical Physics Mathematical Physics Optimization and Control