It is standard to define an induced matrix norm $|||\cdot|||$ from a vector norm $||\cdot||$ in this way:

$|||A|||=\max_{x \neq 0}{\frac{||Ax||}{||x||}}$.

Suppose we define a different function of matrices this way:

$f(A)=\inf_{x \neq 0}{\frac{||Ax||}{||x||}}$.

Has $f(\cdot)$ been studied before? Does it have a standard name?