  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.
