TitleBesicovitch's 12 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
Abstract

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 Ti\v{s}er, showing that the statement is indeed true if 12 is replaced by 710 (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.

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