Analysis Seminar

In 1928 Besicovitch formulated the following conjecture, probably the oldest open problem in geometric measure theory. Let E be a closed subset of the plane with finite length (more precisely finite Hausdorff 1-dimensional measure) and assume its length is more than half of the diameter in all sufficiently small disks centered at a.a. its points. Then E is rectifiable, i.e. it lies in a countable union of C1 arcs with the exception of a set of points with zero length. 1/2 cannot be lowered, while Besicovitch himself showed that the statement holds if it is replaced by 3/4. His bound was improved only once by Preiss and Tiser in the nineties to an (algebraic) number which is approximately 0.735. In this talk I will report on further progress stemming from a joint work with Federico Glaudo, Annalisa Massaccesi, and Davide Vittone. Besides improving the bound of Preiss and Tiser to a substantially lower number (0.7), our work proposes a family of variational methods to find and improve the latter bound.