1.1. LECTUR E 1
THEOREM
1.1 . For every t e R and £ G
S'n_1
w e have
PROOF.
Not e tha t -Af(i ) = fr
x
t\
=tx{\\x\\oo)dx,
wher e x i s the indicato r
function o f the interva l [—1,1] . Takin g the Fourie r transform i n the variabl e
t an d usin g the Fubin i theore m w e get
A^(r) = f A^(t)e-
Ürdt
= [ e~
ür
[ x(\Moo)dx dt =
JR JR J(x£)=t
= [ x(\\x\\oo)e-
ir^dx.
JRn
Clearly, xdklU ) = x(\
xi\)
x ( W ), therefor e
Mr) -flf
1
e-
irX^dxk
= T f[ ^ ^ .
Since the functio n A^(t) i s even, w e get
2*A((t) = (^)
A(t)
=
2n
f e~
ür
f[ ^T^dr.
r£k
D
We no w giv e a n ide a o f the proo f o f K . Ball' s theore m o n th e maxima l
section o f the cube .
THEOREM
1.2. (K. Ball [Bai ] ; For every £ e S
n~l
V o l n - i f Q n n ^ ) ^ /2 ,
TOt/i equalüy for £ = (^= , ^, 0,... , 0).
SKETCH O F PROOF .
Case 1. Suppos e |^ | 4= , fo r som e k. Obviously , th e volum e o f th e
projection o f Q
n
H^
1-
t o the plan e x
k
= 0 does not excee d the volume of th e
projection o f th e whol e cube , whic h i s 1. O n th e othe r han d th e volum e of
this projectio n i s j£/ J (whic h i s the correspondin g cosine ) multiplie d b y th e
volume o f Q
n
n ^
_L.
So ,
1|&I Voi
n
_!(Qnn^).
The assumptio n o n £& yield s the result .
Case 2 . |£& | -4 = fo r al l k. Applyin g Hölder' s inequalit y t o formul a
(1.1) give s
|sin(7rr£fc)|
/
OO
i"
n
•°°fc=i
Kr£k
dr
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