2 1. HYPERPLAN E SECTION S O F £
P
-BALLS
For larg e dimensions , th e sam e effec t ca n b e see n fro m th e followin g
formula, whic h date s bac k t o Laplac e (1812 ) [La] :
Voln_! Q
n
n -
7 =
, . . . , - =
x
= - / y Y .
;
dr .
V y/n V™ ) T T J ^ V r / v ^ /
Here an d i n th e sequel , fo r £ G
*Sn_1
, w e denote th e centra l hyperplan e
section orthogona l t o £ by £\ i. e

= {xeR
n:(x,Z)
= 0}.
In orde r t o se e tha t fo r larg e dimension s th e sectio n orthogona l t o th e
main diagona l i s no t maximal , on e ca n reproduc e a n argumen t appearin g
also in the classica l centra l limi t theore m i n probability , a s follows :
sinx _ x
2
x
4
1 - ^ + ^7
x 3 ! 5 !
and therefor e a s n - ^ o o
Vouf^n^,...,^)1)
~i /
H
(i-^-) dr~
\ V « y/n ) T T J_
y/K
\ 6nJ
1 r ° , /6 "
~ / e
6
dr
= \
—.
T T J-oo V 7 T
But
Voin_! (Q„ n (-^, -4,o,..•
,o)x)
= V2 y^ .
Surprisingly enough , th e proble m o f finding th e maxima l sectio n o f th e
cube remaine d ope n til i 1986, whe n K . Bal l [Bai ] prove d tha t i n al l di -
mensions th e maximu m i s indee d y/2. Th e proo f wa s base d o n th e genera l
formula fo r centra l hyperplan e section s o f th e cub e goin g bac k t o Poly a
(1913) [Pol] :
(Li) voi
n_!(Qnnex)=-/
n
n Sin{r Jk)dr.
Here we assume tha t sin(r£/ c)/r£/c = 1 if £
fc
= 0 .
To prov e thi s formula , w e us e a n argumen t als o appearin g i n [NP] .
Denote b y
2Qn = B^ = {x G
Rn
: ||:r||oo - ma x \x k\ 1}
lkn
the uni t bal l o f the spac e i^o- Let
^ ( t ) = Voi
n
_i(i%n{£-
L
+ i£} )
be th e ( n l)-dimensiona l volum e of the hyperplan e sectio n o f the bal l JB^
perpendicular t o £ G
S71- 1
an d a t distanc e t from th e origin .
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