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

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 points x and y was conjectured to equidistribute its mass over 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 same measure expectation of ^\alpha for any exponent $$\alpha > 1$$ . The kernel represents objective an infinite-dimensional quadratic program. We verify discrete continuous versions milder in a non-empty range \alpha _{\Delta ^d} \ge establish uniqueness resulting maximizer $${\hat{\mu }}$$ up rotation. show no longer when <\alpha ^d}$$ At endpoint =\alpha range, we another $$\mu $$ must also exist is not rotation For version conjecture, “Appendix A” provided Bilyk et al response earlier draft work combines with present improvements yield ^d}<2$$ original $${\alpha ^d}}=1$$ remains open (unless $$d=1$$ However, maximum possible >1$$ rotations maximize aforementioned uniquely sufficiently small ball $$L^\infty -Kantorovich–Rubinstein–Wasserstein metric $$d_\infty from optimal transportation; true mutually absolutely respect but size depends ,d$$ $$\Vert \frac{d {\hat{\mu }}}{d\mu }\Vert _{\infty }$$