Besicovitch's ½ problem investigates the smallest threshold σ guaranteeing rectifiability for a set with Hausdorff 1-dimensional finite measure when the lower density of the set is larger than σ almost everywhere. Besicovitch conjectured that σ = ½ (hence the name of the problem) and proved σ ≤ ¾, then Preiss and Tišer improved the bound to σ ≤ (2+\sqrt{46})/12 ~ 0.73186. In a recent work in collaboration with C. De Lellis, F. Glaudo and D. Vittone, we devise a strategy to improve the bound by means of a hierarchy of variational problems and we reach a proof that σ ≤ 0.7. In this seminar, I will try to explain the fairly intuitive geometric idea behind this strategy and I will try to summarize both the computational obstacles and the intrinsic obstacles that are still in the way.