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

• 成果类型: Article
• 署名作者: Lim, Tongseok; McCann, Robert J.
• 署名单位: Purdue University System; Purdue University; University of Toronto
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-022-01108-1
• 发表日期: 2022
• 页码: 1197-1214
• 关键词: energy

摘要

Among probability measures on d-dimensional real projective space, one which maximizes the expected angle arccos (x/vertical bar x vertical bar . y/vertical bar y vertical bar) between independently drawn projective points x and y was conjectured to equidistribute its mass over the standard Euclidean basis {e(0), e(1), . . . , e(d)) by Fejes Toth (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) (x/vertical bar x vertical bar . y/vertical bar y vertical bar) for any exponent alpha > 1. The kernel arccos(alpha) (x/vertical bar x vertical bar . y/vertical bar y vertical bar) 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 >= 1, and establish uniqueness of the resulting maximizer (mu) over cap up to rotation. We show (mu) over cap no longer maximizes when alpha < a(Delta)d. At the endpoint alpha = a(Delta)d of this range, we show another maximizer mu must also exist which is not a rotation of <(mu)over cap>. 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 (mu) over cap and its rotations maximize the aforementioned expectation uniquely on a sufficiently small ball in the L-infinity-Kantorovich-Rubinstein-Wasserstein metric d(infinity) from optimal transportation; the same is true for any measure mu which is mutually absolutely continuous with respect to (mu) over cap, but the size of the ball depends on alpha, d, and parallel to d (mu) over cap /d mu parallel to infinity.