An afternoon of analysis talks: Giovanni Alberti (University of Pisa)

Dividing a set in half

In this talk I will describe some result about the following elementary problem, of isoperimetric flavor:
Given a set E in R^d with finite volume, is it possible to find an hyperplane P that cuts 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 this question, even in a stronger form, has a positive answer if the dimension d is 3 or higher.But, interestingly enough, our proof breaks down completely in dimension d=2, and we do not know the answer in this case (but we know that the answer is positive if we allow cuts that are not exactly planar, but close to planar). It turns out that this question has some interesting connection with the Kakeya problem.
This is a work in progress with Alan Chang (Princeton University).

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