On maximal hyperplane sections of the unit ball of lnp for p > 2

The maximal hyperplane section of the l n ∞-ball, i.e. of the n-cube, is the one perpen- dicular to 1 √ 2 ( 1 , 1 , 0 , . . . , 0 ) , as shown by Ball. Eskenazis, Nayar and Tkocz extended this result to the l n p -balls for very large p ≥ 10 15 . By Oleszkiewicz, Ball’s result does not transfer to l n p for 2 < p < p0 26 . 265. Then the hyperplane section perpendic- ular to the main diagonal yields a counterexample for large dimensions n. Suppose that p0 ≤ p < ∞ . We show that the analogue of Ball’s result holds in l n p -balls for all hyperplanes with normal unit vectors a, if all coordinates of a have modulus ≤ 1 √ 2 and p has distance ≥ 2 − p to the even integers. Under similar assumptions, we give a Gaussian upper bound for 20 < p < p0.

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