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 $$, where the endpoints of the chords are drawn according to a given probability distribution on $$. We show that, for $$, the problem is degenerated in the sense that any continuous measure is an extremizer, and that, for $$ sufficiently close to 1, the desired maximal value is strictly below the one for $$ by a polynomial factor in $$. Finally, we prove, by considering the auxiliary problem of drawing a single random chord, that the desired maximum is $$ for $$. Connections with other variational problems and energy minimization problems are also presented.