On hyperplane sections and projections in lnp

For 2 < p < p0 26 . 265, the hyperplane section of the l n p -unit ball B n p perpendicular to a ( n ) = 1 √ n ( 1 , . . . , 1 ) for large n has larger volume than the one orthogonal to a ( 2 ) = 1 √ 2 ( 1 , 1 , 0 , . . . , 0 ) , as shown by Oleszkiewicz. This is different from the case of l n ∞ considered by Ball. We give a quantitative estimate for which dimensions n this happens, namely for n > c ( 1 p0 − p

1 p − 2 ) for some absolute constant c > 0. Correspondingly for projections of B n q onto hyperplanes, Barthe and Naor showed that projections onto hyperplanes perpendicular to a ( n ) have smaller volume for large n than onto the one orthogonal to a ( 2 ) , if 4 3 < q < 2, different from the case q = 1. We show that this happens for all n > 5 ( 1 q − 4 3

1 2 − q ) .

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