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).