I considered a minimization version of matrix p-norms, defined for a matrix $A$ by
$$ f_p(A)= \min_{x\neq 0} \frac{||Ax||_p}{||x||_p}. $$
Notice that $f_p(A) = 0$ if and only if $A$'s columns are linearly dependent. Therefore $f_p(A)$ can be seen as a measure of the linear independence of $A$'s columns: the larger $f_p(A)$ is, the more independent $A$'s columns are.
When $p=2$, $f_2(A)$ equals the smallest singular value of $A$ and can be computed by doing SVD. When $p\neq 2$, is there any known algorithm than can compute $f_p(A)$? Or if this problem has not been studied before for the $p\neq 2$ case, is it likely to be of interest?