Besicovitch's 1/2 problem and linear programming
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913360754245632 |
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| author | De Lellis, Camillo Glaudo, Federico Massaccesi, Annalisa Vittone, Davide |
| author_facet | De Lellis, Camillo Glaudo, Federico Massaccesi, Annalisa Vittone, Davide |
| contents | 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 Tišer, showing that the statement is indeed true if $\frac{1}{2}$ is replaced by $\frac{7}{10}$ (in fact we improve the Preiss-Tišer 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_17536 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Besicovitch's 1/2 problem and linear programming De Lellis, Camillo Glaudo, Federico Massaccesi, Annalisa Vittone, Davide Classical Analysis and ODEs Analysis of PDEs Metric Geometry 28A75, 49Q15, 90C05, 68V05 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 Tišer, showing that the statement is indeed true if $\frac{1}{2}$ is replaced by $\frac{7}{10}$ (in fact we improve the Preiss-Tišer 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. |
| title | Besicovitch's 1/2 problem and linear programming |
| topic | Classical Analysis and ODEs Analysis of PDEs Metric Geometry 28A75, 49Q15, 90C05, 68V05 |
| url | https://arxiv.org/abs/2404.17536 |