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arXiv · 2404.17536

Besicovitch's 1/2 problem and linear programming

Camillo De Lellis·Federico Glaudo·Annalisa Massaccesi·Davide Vittone
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Abstract

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.

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Camillo De Lellis, Federico Glaudo, Annalisa Massaccesi, Davide Vittone. 2024-04-26. Besicovitch's 1/2 problem and linear programming. https://arxiv.org/abs/2404.17536

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