Mathematical analysis reveals bounded Buffon discrepancy for line intersections in convex sets, demonstrating optimal geometric distributions using Steinhaus constructions.
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 .
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Stefan Steinerberger (2026) studied this question.
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