CHAPTER 1
Hyperplane section s o f £ p-balls
1.1. Lectur e 1
Historically, th e Fourie r analyti c approac h t o section s o f conve x bodie s
originates fro m th e stud y o f hyperplan e section s o f th e uni t cub e Q
n
=
[—1/2,
l/2]n
i n R
n.
Lookin g a t th e thre e dimensiona l cube , on e ma y thin k
that th e maxima l hyperplan e sectio n o f th e cub e i s th e on e perpendicula r
to th e mai n diagonal :
Figure 1. Sectio n orthogona l t o th e mai n diagonal .
However, a quick computation show s that th e are a of this section is ^ p ,
which is smaller than \/2 , the area of the section perpendicular t o the vecto r
(1,1,0):
Figure 2 . Sectio n orthogona l t o th e vecto r (1,1,0).
i
http://dx.doi.org/10.1090/cbms/108/01
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