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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