On hyperplane sections and projections in lₚⁿ

For 2< p < p₀ 26.265 2 < p < p 0 ≃ 26.265 , the hyperplane section of the lₚⁿ l p n -unit ball Bₚⁿ B p n perpendicular to a⁽ⁿ⁾ = 1√n (1 ,… ,1) 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) a ( 2 ) = 1 2 ( 1 , 1 , 0 , … , 0 ) , as shown by Oleszkiewicz. This is different from the case of l_∞ ⁿ 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) n > c ( 1 p 0 - p + 1 p - 2 ) for some absolute constant $$c>0$$ c > 0 . Correspondingly for projections of Bqⁿ B q n onto hyperplanes, Barthe and Naor showed that projections onto hyperplanes perpendicular to a⁽ⁿ⁾ a ( n ) have smaller volume for large n than onto the one orthogonal to a⁽²⁾ a ( 2 ) , if 4/3< q < 2 4 3 < q < 2 , different from the case $$q=1$$ q = 1 . We show that this happens for all n > 5 (1q-4/3 + 1/2-q) n > 5 ( 1 q - 4 3 + 1 2 - q ) .