Camillo De Lellis: Besicovitch's 1/2 problem and linear programming

Seminars - Analysis and Applied Mathematics Seminar
CAMILLO DE LELLIS, Institute for Advanced Studies
12:30 - 13:45
room 3-E4-SR03 (Rontgen)


In 1928 Besicovitch formulated the following conjecture. Let E be a Borel subset of the plane with finite length 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 C^1 arcs with the exception of a null set. 1/2 cannot be lowered, while Besicovitch himself showed that the statment holds if it is replaced by 3/4. His bound was improved only once by Preiss and Tiser in the nineties to a 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, our work uncovers an interesting connection with a class of linear programming problems.


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