Buffon discrepancy and the Steinhaus longimeter

Abstract

Abstract Let be a convex set. We study the problem of distributing a one‐dimensional set with total length so that for any line in the number of intersections is proportional to the length as much as possible; we use the term Buffon discrepancy for the largest error. A construction of Steinhaus can be generalized to prove the existence of sets with Buffon discrepancy . We also show that the unit disk admits a set with uniformly bounded Buffon discrepancy as .