Buffon Discrepancy and the Steinhaus Longimeter

Let $\mathrm{\Omega }\subset {\mathbb{R}}^2$ be a convex set. We study the problem of distributing a one-dimensional set $S$ with total length $L$ so that for any line $\mathrm{\ell }$ in ${\mathbb{R}}^2$ the number of intersections $\mathrm{\#}(\mathrm{\ell }\cap S)$ is proportional to the length ${\mathcal{H}}^1(\mathrm{\ell }\cap \mathrm{\Omega })$ 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 $\lesssim L^{1\mathrm{/}3}$. We also show that the unit disk $\mathbb{D}$ admits a set with uniformly bounded Buffon discrepancy as $L\to \mathrm{\infty }$.