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 1/2 almost everywhere must be countably rectifiable. We improve the best known bound, due to Preiss and Ti{s}er, showing that the statement is indeed true if 1/2 is replaced by 7/10 (in fact we improve the Preiss-Ti{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.