On hyperplane sections and projections in lₚⁿ

For 2 < p < p₀ 26.265, the hyperplane section of the lₚⁿ-unit ball Bₚⁿ perpendicular to a^(n) = 1/sqrt(n) (1, ... ,1) for large n has larger volume than the one orthogonal to a^(2) = 1/sqrt(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 Bqⁿ 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).