On hyperplane sections and projections in $$l_p^n$$

Abstract

AbstractFor $$2< p < p_0 \simeq 26.265$$ 2 < p < p 0 ≃ 26.265 , the hyperplane section of the $$l_p^n$$ l p n -unit ball $$B_p^n$$ B p n perpendicular to $$a^{(n)} = \frac{1}{\sqrt{n}} (1 ,\ldots ,1)$$ a ( n ) = 1 n ( 1 , … , 1 ) for large n has larger volume than the one orthogonal to $$a^{(2)} = \frac{1}{\sqrt{2}} (1,1,0 ,\ldots ,0)$$ a ( 2 ) = 1 2 ( 1 , 1 , 0 , … , 0 ) , as shown by Oleszkiewicz. This is different from the case of $$l_\infty ^n$$ l ∞ n considered by Ball. We give a quantitative estimate for which dimensions n this happens, namely for $$n > c (\frac{1}{p_0-p} + \frac{1}{p-2})$$ n > c ( 1 p 0 - p + 1 p - 2 ) for some absolute constant $$c>0$$ c > 0 . Correspondingly for projections of $$B_q^n$$ 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 $$\frac{4}{3}< q < 2$$ 4 3 < q < 2 , different from the case $$q=1$$ q = 1 . We show that this happens for all $$n > 5 (\frac{1}{q-\frac{4}{3}} + \frac{1}{2-q})$$ n > 5 ( 1 q - 4 3 + 1 2 - q ) .