In a few questions for upper bounds of the norm of solutions to some bilinear systems, we are often interested in the problem of the "minimal norm of a linear map": $$ g(A) = \inf_{\|x\| = 1} \|Ax\|, $$ for some norm $\|\cdot\|$.
Is there a name for such a quantity, and are there references for it? Since the problem is nonconvex (and does not appear to have a natural relaxation or rewriting to make it so), it seems to be very hard in practice to solve exactly for arbitrary norms (but it is possible I may be wrong).
There do appear to be some easy cases. For example, when the norm is the usual Euclidean norm (e.g., $\|x\|_2 = \sqrt{x^Tx}$), the solution is clearly the smallest singular value of $A$, but other norms do not really seem to appear in many places (though, it's quite likely my Google-fu is failing me!).
Additionally, there is some interest in somewhat-tight lower bounds for these quantities, which would be very useful.
Here is a trick for square matrices. If $A$ fails to be invertible, then of course the infimum is zero. In the case that $A$ is invertible, we have $$ \begin{align} \inf_{\|x\| = 1} \|Ax\| &= \inf_{x \neq 0} \frac{\|Ax\|}{\|x\|} = \inf_{y\neq 0}\frac{\|A(A^{-1}y)\|}{\|A^{-1}y\|} \\ & = \inf_{y \neq 0}\frac{\|y\|}{\|A^{-1}y\|} = \inf_{\|y\| = 1}\frac{1}{\|A^{-1}y\|} = \left[\sup_{\|y\| = 1} \|A^{-1}y\|\right]^{-1}. \end{align} $$ A lower bound in the case that $A$ is rectangular. If $A$ fails to have full column rank, then the infimum must be zero. If $A$ does have full column rank, let $B$ be such that $BA = I$. We have $$ \begin{align} \inf_{\|x\| = 1} \|Ax\| &= \inf_{x \neq 0} \frac{\|Ax\|}{\|x\|} = \inf_{x \neq 0} \frac{\|(Ax)\|}{\|B(Ax)\|} \\ & \geq \inf_{y\neq 0}\frac{\|y\|}{\|By\|} = \inf_{\|y\| = 1}\frac{1}{\|By\|} = \left[\sup_{\|y\| = 1} \|By\|\right]^{-1}. \end{align} $$