Some sharp inequalities for norms in \(\mathbb {R}^n\) and \(\mathbb {C}^n\)

The main result of this paper is that for any norm on a complex or real n-dimensional linear space, every extremal basis satisfies inverted triangle inequality with scaling factor \(2^n-1\) $2^n-1$ . Furthermore, the constant \(2^n-1\) $2^n-1$ is tight. We also prove that the norms of any two extremal bases are comparable with a factor of \(2^n-1\) $2^n-1$ , which, intuitively, means that any two extremal bases are quantitatively equivalent with the stated tolerance.