On the complex Banach conjecture

Abstract

The complex conjecture of Stefan Banach states that if V is a Banach space over the complex numbers where for some n, 1 < n < dim(V ), all of its subspaces of dimension n are isometric, then V is a Hilbert space. Mikhail Gromov proved it for n even. Here, we prove it for n ≡ 1 mod 4.

Extremal inscribed and circumscribed complex ellipsoids

“…Observe that all complex ellipsoids are real but not the other way around. The difference between real and complex ellipsoids is clearly explained by the following result from [2].…”