2020/07/26 by Tongseok Lim, Robert J. McCann, Lim, Tongseok +1
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #FOS: Mathematics #FOS: Physical sciences #Mathematical Approximation and Integration #Mathematical Physics (math-ph) #Metric Geometry (math.MG) #Optimization and Control (math.OC) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2007.13052
openalex publication_date 2020/07/26 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
Among probability measures on d-dimensional real projective space, one which maximizes the expected angle \arccos((x)/(|x|)⋅ (y)/(|y|)) between independently drawn projective points x and y was conjectured to equidistribute its mass over the standard Euclidean basis \e0,e1,…, ed\ by Fejes Tóth \citeFT59. If true, this conjecture evidently implies the same measure maximizes the expectation of \arccosα((x)/(|x|)⋅ (y)/(|y|)) for any exponent α> 1. The kernel \arccosα((x)/(|x|)⋅ (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> αΔd ≥ 1, and establish uniqueness of the resulting maximizer μ up to rotation. We show μ no longer maximizes when α1, we show μ and its rotations maximize the aforementioned expectation uniquely on a sufficiently small ball in the L^∞-Kantorovich-Rubinstein-Wasserstein metric d_∞ from optimal transportation; the same is true for any measure μ which is mutually absolutely continuous with respect to μ, but the size of the ball depends on α,d, and ‖(d μ)/(dμ)‖∞.