Conference on Calculus of Variations in Lille - 3rd edition - July 4-6 2022

Dividing a set in half

Jul 4, 2022, 11:30 AM

1h

M2 building, Cité Scientifique - Meeting room, 1st floor (Laboratoire Paul Painlevé)

Speaker

Giovanni Alberti (Università di Pisa)

Description

In this talk I will consider the following problem of isoperimetric type:

Given a set E in ${\mathbb{R}}^d$ with finite volume, is it possible to find an hyperplane $P$ that splits $E$ in two parts with equal volume, and such that the area of the cut (that is, the intersection of $P$ and $E$) is of the expected order, namely $(vol(E){)}^{1-1/d}$?

We can show that the answer is positive if the dimension $d$ is $3$ or higher, but, somewhat surprisingly, our proof breaks down completely in dimension $d=2$, and we do not know what happens in this case.
(However we know that the answer is positive even for $d=2$ if we allow cuts that are not exactly planar, but close to planar.)

This is a work in progress with Alan Chang (Princeton University).