Besicovitch’s 12 problem and linear programming

We consider the following classical conjecture of Besicovitch: a 1 1 -dimensional Borel set in the plane with finite Hausdorff 1 1 -dimensional measure H 1 H¹ which has lower density strictly larger than 1 2 12 almost everywhere must be countably rectifiable. We improve the best known bound, due to Schechter, showing that the statement is indeed true if 1 2 12 is replaced by 7 10 710. Concerning Schechter’s bound, which is in fact an improvement of an argument by Preiss and Tišer, and it is valid in any metric space, we give a simpler proof. 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.