TitleBesicovitch's $\frac{1}{2}$ problem and linear programming
Publication TypeJournal Article
Year of Publication2024
AuthorsDe Lellis C, Glaudo F, Massaccesi A, Vittone D
Type of Articlegeometric measure theory
KeywordsBesicovitch conjecture, linear programming, rectifiable sets

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.