Camillo De Lellis – Institute for Advanced Study
Wednesday, May 21st, 2025
Zoom opens: 10:30AM DST
Seminar Begins: 10:45AM DST
Besicovitch's 1/2 problem and linear programming
In 1928 Besicovitch formulated the following conjecture. Let E be a 1-dimensional set of the plane. The set E cannot be a fractal if at all sufficiently small scale around (almost all) its points the length of the set in a disk is slightly more than half of the diameter of the disk. It was proved by Dickinson that the threshold cannot be lowered, while Besicovitch himself showed first that the statement holds if the thresold is slightly smaller than 1, and improved it in 1938 to 3/4. Since then 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, our work proposes a family of variational methods to find and improve the latter bound. We can improve Preiss and Tiser bounds both with a pen-and-paper proof and with the assistance of a computer (which is used to examine a very large, but finite, number of cases). The latter is in fact a feasible computation because a part of the variational problems can be formulated as a linear programming task.