Complex Hanner's Inequality for Many Functions

Abstract

We establish Hanner's inequality for arbitrarily many functions in the setting where the Rademacher distribution is replaced with higher dimensional random vectors uniform on Euclidean spheres.

Stability of polydisc slicing

“…Across the areas in convex geometry, significant efforts have been made to extend many fundamental and classical results well-known from real spaces to complex ones. For example, see [3,7,9,11,12,18,19,21,30,31,[33][34][35] (sometimes complex-counterparts turn out to be "easier," e.g., [28,39,46], but for certain problems, on the contrary, satisfactory results have been elusive, e.g., [48]). A counterpart of Ball's cube slicing in ℂ 𝑛 was discovered by Oleszkiewicz and Pełczyński in [42].…”