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SUMMARY:An afternoon of analysis talks: Giovanni Alberti (University of Pisa)
LOCATION:Spiegelgasse 1, Seminarraum 00.003
DESCRIPTION:In this talk I will describe some result about the following el
ementary problem, of isoperimetric flavor: Given a set E in R^d
with finite volume, is it possible to find an hyperplane P tha
t 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 ex
pected 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 proo
f breaks down completely in dimension d=2, and we do not know t
he answer in this case (but we know that the answer is positive
if we allow cuts that are not exactly planar, but close to pla
nar). It turns out that this question has some interesting con
nection with the Kakeya problem. This is a work in progress wit
h Alan Chang (Princeton University).
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