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 ∈ (0,1], where the endpoints of the chords are drawn according to a given probability distribution on S¹. 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.