Hausdorff measures in the Bertrand's random chord problem

TL;DR

This paper introduces a novel metric space framework using Hausdorff measures to analyze the Bertrand's chord problem, providing new insights into the distribution and dimensions of chords.

Contribution

It develops a new metric space approach with Hausdorff measures for the Bertrand problem, offering a fresh solution and detailed geometric analysis.

Findings

Established the form of open and closed balls in the metric space

Computed Hausdorff dimensions of these balls

Provided a new solution to the Bertrand problem using a continuous uniform distribution analogue

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.