Calculus of Variations and Geometric Measure Theory

C. De Lellis - F. Glaudo - A. Massaccesi - D. Vittone

Besicovitch's $\frac{1}{2}$ problem and linear programming

created by delellis on 25 Apr 2024
modified on 23 May 2024

[BibTeX]

Preprint

Inserted: 25 apr 2024
Last Updated: 23 may 2024

Year: 2024

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.

Keywords: Rectifiability, Besicovitch conjecture, Linear programming


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