2024/04/26 by Camillo De Lellis, De Lellis, Camillo, Federico Glaudo +5
Engineering · Mathematics · #28A75 #49Q15 #68V05 #90C05 #Advanced Optimization Algorithms Research #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Metric Geometry (math.MG) #Optimization and Mathematical Programming #Scheduling and Optimization Algorithms
paper · pdf · doi:10.48550/arxiv.2404.17536
openalex publication_date 2024/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (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 (1)/(2) is replaced by (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.