Title | Besicovitch's 12 problem and linear programming |
Publication Type | Journal Article |
Year of Publication | 2024 |
Authors | De Lellis C, Glaudo F, Massaccesi A, Vittone D |
Type of Article | geometric measure theory |
Keywords | Besicovitch 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. |
Notes | Preprint |
Order:
13