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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 triangleinequality with scaling factor $2^n-1$. Furthermore, the constant $2^n-1$ istight. We also prove that the norms of any two extremal bases are comparablewith a factor of $2^n-1$, which, intuitively, means that any two extremal basesare quantitatively equivalent with the stated tolerance.

Some inequalities for norms in $\mathbb{ R}^n$ and $\mathbb{ C}^n$