Abstract
Abstract The problem of maximal hyperplane section of B p ( C n ) with p ≥ 1 is considered, which is the complex version of central hyperplane section problem of B p ( R n ) . The relation between the complex slicing problem and the complex isotropic constant of a body is established, an upper bound estimate for the volume of complex central hyperplane sections of normalized complex ℓ p ( C n ) -balls that does not depend on n and p is shown, which extends results of Oleszkiewicz and Pełczyński, Koldobsky and Zymonopoulou, and Meyer and Pajor. MSC:52A21, 46B07.
Highlights
Let Bp(Rn) and Bp(Cn) denote the unit balls of the real and complex n-dimensional p spaces, p(Rn) and p(Cn), respectively
We identify Cn with R n using the standard mapping, that is, for ξ = (ξ, . . . ξn) = (ξ + iξ, . . . , ξn + iξn ) ∈ Cn, (ξ + iξ, . . . , ξn + iξn ) →τ (ξ, ξ, . . . , ξn, ξn )
Since norms on Cn satisfy the equality λz = |λ| z, ∀z ∈ Cn, ∀λ ∈ C, origin-symmetric complex convex bodies correspond to those origin-symmetric convex bodies K in R n that are invariant with respect to any coordinate-wise two-dimensional rotation, namely for each θ ∈ [, π ] and each (ξ, ξ, . . . ξn, ξn ) ∈ R n ξ K = Rθ (ξ, ξ ), . . . , Rθ K, ( . )
Summary
Let Bp(Rn) and Bp(Cn) denote the unit balls of the real and complex n-dimensional p spaces, p(Rn) and p(Cn), respectively. The extremal volume of central hyperplane section of Bp(Rn) is studied by various authors (see, e.g., [ – ]). The right-hand side of the following inequalities is due to Ma and the third named author [ ], and it shows an upper bound estimate for the volume of central hyperplane sections of normalized p-balls that does not depend on n and p.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
