Let $X$ be a finite dimensional Banach space and define a matrix norm $\| \cdot \|_{X}$ by

$$ \| A \|_{X} = \sup_{x \ne 0} \frac{\|A x\|_{X}}{\|x\|_{X}} $$

where the matrix $A$ is interpreted as an operator $X \to X$ in the obvious way.

I'm looking to define a minimal norm, namely

$$ f(A) = \inf_{X} \| A \|_{X} $$

What can be said about $f$? Does it define a norm? Can it be computed?