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

Monday 29 Jul 2024, 15:30 → 16:30 Europe/Rome

Rectorate/Building-Auditorium (GSSI)

In 1928 Besicovitch formulated the following conjecture, which is probably the oldest open problem in geometric measure theory. Assume E is a closed 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 almost all its points. Then E cannot be a ``fractal''. More precisely 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 conclusion 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. Our sharpest result is that Besicovitch's conjecture is correct if we replace 1/2 with 7/10, but besides giving a substantially lower number than what was known before, our work uncovers an interesting connection with a class of linear programming problems, and provides a variational method to compute much better bounds. 

Link Zoom:
https://us02web.zoom.us/j/86086663267?pwd=ZzRBNTI0bUIzTU9ReHVtdlhXblljZz09

ID riunione: 860 8666 3267
Codice d’accesso: 965677