On Extremal Problems Associated with Random Chords on a Circle

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.