Hausdorff measures in the Bertrand's random chord problem

Abstract

A set of chords of a circle of given radius is represented as a metric space w.r.t. a metric introduced by Hausdorf. The form of open and closed balls with respect to this metric is established. We consider a family of Hausdorff outer measures generated by this metric. We compute the Hausdorff dimension of open and closed balls. An analogue of a continuous uniform distribution is introduced and a new solution of the Bertrand problem is given with an old answer.