On Extremal Problems Associated with Random Chords on a Circle

TL;DR

This paper investigates extremal probabilities of intersecting two random chords in a circle, revealing degenerate cases at full radius and specific maximum probabilities for smaller radii, with connections to variational problems.

Contribution

It characterizes extremizers for the probability of intersecting chords in a circle, showing degeneracy at radius one and establishing maximum probabilities for smaller radii.

Findings

Any continuous measure is an extremizer at radius 1.

Maximum probability is 1/4 for radius less than 1/2.

Maximum probability decreases polynomially as radius approaches 1.

Abstract

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 $r, \, r \in (0,1]$, where the endpoints of the chords are drawn according to a given probability distribution on $\mathbb{S}^1$. 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 $r \in (0,1/2).$ Connections with other variational problems and energy minimization problems are also presented.