k=l
oo
-oo
k 1
sin(7rrffc)
nr£k
/»oo
J oo
l
fc
r
L
sin(7rx)
7TX
^k
dr \ =
i
72"
£2
Sfc
(ix
The mai n ingredien t o f th e proo f i s Ball' s integra l inequality , whic h say s
that fo r al l s 2 ,
f
J
sin(7rx)
7TX
dx \ I - .
The proof o f this inequality ics quite difficult an d ca n be found i n the origina l
paper o f Ball or in the pape r [NP ] (th e latter proo f wa s reproduced i n [K9 ,
p. 145]). Applyin g Ball' s integra l inequalit y wit h s JJ (not e tha t s 2
^k
because o f the assumptio n £* . -4=) , we get
n , v ^ £ 2
n
D
Let u s no w conside r th e uni t ball s B™ o f th e Space s £™, 0 q oo,
defined b y
K =
{xe
\x
= (\x
1
\i + --- + \x n\i)1/il}.
Meyer an d Pajo r [MeyP ] discovere d a n analo g o f formul a (1.1 ) fo r centra l
hyperplane section s o f the ball s B™. Th e origina l resul t o f Meyer an d Pajo r
was proved i n the cas e 1 q 2 using probabihsti c methods , an d late r th e
formula wa s extended t o al l 0 q oc in [K4 ] b y Fourie r methods .
(1.2)
THEOREM 1.3. For every £ G Sn l ,
Vol
n
_i(ß"n£-
L
) =
7r(n - l)r (
/»OO
,L
T")h
k=i
where'yq(t) = (e-W
i)A(t),
te
Theorem 1.3 wil l b e prove d i n Lectur e 2 a s a par t o f a mor e genera l
result. No w le t u s sho w a n applicatio n o f thi s theore m t o th e proble m o f
finding th e extrema l centra l section s o f ^-balls , 0 q 2 . W e star t wit h
some properties of the functions j
q
(for more about thes e functions, se e [K9,
Section 2.8]) .
LEMMA
1.4. For 0 q 2, the function ^
q
is positive ort [0, oo), and
the function \og(
/yq(y/x))
is convex on [0 , oo).
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