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Buffon Discrepancy and the Steinhaus Longimeter

Stefan Steinerberger

Abstract

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

Buffon Discrepancy and the Steinhaus Longimeter

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 .

Paper Structure

This paper contains 11 sections, 3 theorems, 38 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The problem
  3. Buffon discrepancy
  4. Unit Disk
  5. The Steinhaus longimeter
  6. Open problems.
  7. Related results.
  8. Proofs
  9. Proof of the Proposition.
  10. Proof of Theorem 1
  11. Proof of Theorem 2

Key Result

Proposition 1

Let $\Omega \subset \mathbb{R}^2$ be bounded, let $S$ be a set with length $\mathcal{H}^1(S) = L$ having the property that for some $c>0$ and all lines $\ell$ in $\mathbb{R}^2$ then

Figures (8)

  • Figure 1: Sets of lines of length $L =500$ inside the unit disk (left) and the Reuleaux triangle (right) with the property that every line through the domain intersects them a number of times roughly proportional to the length (with a fairly small error).
  • Figure 2: A set with length $L=500$ in the unit disk and a line showing that the Buffon discrepancy of the set is $\geq 19.24$.
  • Figure 3: Two sets of line segments with total length $L=500$ inside the unit disk with very small Buffon discrepancy.
  • Figure 4: The sets $S_{n,1/5}$ for $1 \leq n \leq 5$.
  • Figure 5: Construction in the style of the Steinhaus longimeter.
  • ...and 3 more figures

Theorems & Definitions (7)

  • Proposition : Cauchy-Crofton scaling
  • Definition
  • Theorem 1
  • Theorem 2
  • proof
  • proof
  • proof