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 \(2^n-1\). Furthermore, the constant \(2^n-1\) is tight. We also prove that the norms of any two extremal bases are comparable with a factor of \(2^n-1\), which, intuitively, means that any two extremal bases are quantitatively equivalent with the stated tolerance.