Hausdorff dimension of union of lines that cover a curve

TL;DR

This paper explores the Hausdorff dimension of unions of lines covering various types of curves, demonstrating that differentiability imposes strict limitations on the dimension of such unions.

Contribution

It constructs specific curves with line unions of Hausdorff dimension 1 and proves dimension constraints for twice differentiable curves, advancing understanding of geometric covering properties.

Findings

Constructed a differentiable curve with line union Hausdorff dimension 1.

Proved that twice differentiable curves' line unions must have Hausdorff dimension 2.

Showed tangent lines of convex functions can have union dimension 1.

Abstract

We construct a continuously differentiable curve in the plane that can be covered by a collection of lines such that every line intersects the curve at a single point and the union of the lines has Hausdorff dimension 1. We show that for twice differentiable curves this is impossible. In that case, the union of the lines must have Hausdorff dimension 2. If we use only tangent lines then the differentiability of the curve already implies that the union of the lines must have Hausdorff dimension 2, unless the curve is a line. We also construct a continuous curve, which is in fact the graph of a strictly convex function, such that the union of (one sided) tangent lines has Hausdorff dimension 1.