Besicovitch's 1/2 problem and linear programming

TL;DR

This paper advances the understanding of Besicovitch's 1/2 conjecture by improving the density bound to 7/10 and introduces variational problems to explore further bounds and properties related to rectifiability.

Contribution

It improves the known density bound in Besicovitch's conjecture from 1/2 to 7/10 and introduces a family of variational problems to analyze and potentially improve these bounds.

Findings

Bound improved to 7/10 for the conjecture.

Proposed variational problems for further bounds.

Studied properties of these variational problems.

Abstract

We consider the following classical conjecture of Besicovitch: a $1$-dimensional Borel set in the plane with finite Hausdorff $1$-dimensional measure $\mathcal{H}^1$ which has lower density strictly larger than $\frac{1}{2}$ almost everywhere must be countably rectifiable. We improve the best known bound, due to Preiss and Ti\v{s}er, showing that the statement is indeed true if $\frac{1}{2}$ is replaced by $\frac{7}{10}$ (in fact we improve the Preiss-Ti\v{s}er bound even for the corresponding statement in general metric spaces). More importantly, we propose a family of variational problems to produce the latter and many other similar bounds and we study several properties of them, paving the way for further improvements.