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 (Acta Math Acad Sci Hung 10:13–19, 1959. https://doi.org/10.1007/BF02063286). 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 A” 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 ${\alpha _{\Delta ^d}}=1$ 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 $\Vert \frac{d {\hat{\mu }}}{d\mu }\Vert _{\infty }$.

Appendix A. By Dmitriy Bilyk, Alexey Glazyrin, Ryan Matzke, Josiah Park, and Oleksandr Vlasiuk

Theorem A.1

(Estimating the threshold exponent: ${\alpha _{\Delta ^d}}\le 2$) For every $\alpha \ge 2$, the set of maximizers of (1.5) is precisely $\mathcal {P}_{{\Delta }}^=(\mathbf{S}^d)$.

Proof

For $t \in [-1,1]$ set $f_\alpha (t) = (\frac{2}{\pi } \arccos |t| )^\alpha$ and $g(t)=1-t^2$. We claim

$$\begin{aligned} g \ge f_\alpha \text { on } [-1,1] \text { and } \{g=f_\alpha \} = \{-1,0,1\} \iff \alpha \ge 2. \end{aligned}$$

(A.1)

Indeed, setting $h_\alpha (t) := g(t)^{1/\alpha } - f_\alpha (t)^{1/\alpha }$, the computation $h_2''(t) = (2t\pi ^{-1} -1)(1-t^2)^{-3/2}$ shows $h_2(t)$ to be strictly concave on the interval $t \in [0,1]$ and to vanish at both endpoints. For $\alpha = 2$ this establishes (A.1). For $\alpha >2$, the same conclusion then follows from $\frac{dh_\alpha }{d\alpha }(t)\ge 0$. For $\alpha <2$ we have $\lim _{t \nearrow 1} f_\alpha '(t)=- \infty <g'(1)$ and $f_\alpha (1)=0=g(1)$, thus domination of $f_\alpha$ by g fails, confirming the reverse implication in (A.1).

Now (1.4) may be rewritten in the form $E_\alpha (\mu ) = {F_{f_\alpha }}(\mu )$ where

$$\begin{aligned} {F_f}(\mu ) := \iint f(x \cdot y) d\mu (x)d\mu (y). \end{aligned}$$

Let $\mu \in \mathcal {P}(\mathbf{S}^d)$, $\sigma$ be the uniform probability on $\mathbf{S}^d$, and ${\hat{\mu }} \in \mathcal {P}_{{\Delta }}^=(\mathbf{S}^d)$. For $\alpha \ge 2$ we claim

$$\begin{aligned} {F_{f_\alpha }}(\mu ) \le {F_{g}}(\mu ) \le {F_g}(\sigma ) = {F_g}(\hat{\mu }) = {F_{f_\alpha }} ({\hat{\mu }}) \end{aligned}$$

(A.2)

where the middle two (in)equalities for g are known and reproved below, while the first and last follow from (A.1)—which also makes the first inequality strict unless $\mu$ lies in the narrow closure of $\mathcal {P}_{{\Delta }}(\mathbf{S}^d)$. On the other hand, for $\mu \in \overline{\mathcal {P}_{{\Delta }}(\mathbf{S}^d)}$, (1.7) implies that the second inequality in (A.2) becomes strict unless $\mu \in \mathcal {P}_{{\Delta }}^=(\mathbf{S}^d)$.

It remains to establish the middle two (in)equalities in (A.2) for g, which can be done in various ways, c.f. [4]. For example, defining the symmetric $(d+1) \times (d+1)$ matrix $I(\mu )$ by

$$\begin{aligned} I^{ij}(\mu ) = \int _{\mathbf{S}^d} x^i x^j d\mu (x), \end{aligned}$$

the Cauchy–Schwartz inequality for the Hilbert-Schmidt norm yields

since on $\mathbf{S}^d$. If $I(\mu )$ is a multiple of the identity matrix—as for $\mu \in \{\hat{\mu },\sigma \}$—then equality holds. This establishes (A.2) and completes the proof. $\square$