2024/06/10 by Cynthia Bortolotto, João P. G. Ramos, Bortolotto, Cynthia +1
Mathematics · #FOS: Mathematics #Mathematical Approximation and Integration #Metric Geometry (math.MG) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2406.06771
openalex publication_date 2024/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Inspired by the work of Karamata, we consider an extremization problem associated with the probability of intersecting two random chords inside a circle of radius r, r ∈ (0,1], where the endpoints of the chords are drawn according to a given probability distribution on \mathbbS1. We show that, for r=1, the problem is degenerated in the sense that any continuous measure is an extremiser, and that, for r sufficiently close to 1, the desired maximal value is strictly below the one for r=1 by a polynomial factor in 1-r. Finally, we prove, by considering the auxiliary problem of drawing a single random chord, that the desired maximum is 1/4 for r ∈ (0,1/2). Connections with other variational problems and energy minimization problems are also presented.