Besicovitch's 1/2 problem and linear programming

We consider the following classical conjecture of Besicovitch: a 1-dimensional Borel set in the plane with finite Hausdorff 1-dimensional measure H¹ which has lower density strictly larger than 12 almost everywhere must be countably rectifiable. We improve the best known bound, due to Preiss and Tiser, showing that the statement is indeed true if 12 is replaced by 710 (in fact we improve the Preiss-Tiser 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.