Is it possible to construct a matrix norm that uses minimum instead of maximum over a compact convex set?

Question

I'm reading a paper where the following matrix norm is used:

$$ ||A||_{C, 2} = \max_{x \in C} \|Ax\|_2, $$

where A is a $d \times q $ matrix, $C$ is a compact convex set in $\mathbb{R}^q$, and $\|.\|_2$is a standard Euclidean norm in $\mathbb{R}^d$.

The paper uses this norm to prove some properties of some function $\Phi(A)$ over different possible matrices $A$. I want to extend the result of paper in such a manner that the following quantity arises:

$$ s_1(A) = \min_{x \in C} \|Ax\|_2,$$

that is, I need to operate with minimum instead of maximum in all proofs of that paper.

So, the question is as follows: is it possible to define such a norm that will operate with a minimum instead of maximum?

By itself, $s_1(A)$ is not a norm because it is not sub-additive. The following function is sub-additive:

$$ s_2(A) = \frac{1}{\min_{x \in C} \|Ax\|_2},$$

but - another drawback - it causes that $C$ should not contain $x = 0$. To overcome this, we can define the following function:

$$ s_3(A) = \frac{1}{\min_{x \in C \setminus B_{\varepsilon}(0)} \|Ax\|_2},$$

where $B_{\varepsilon}(0)$ is an open ball centered at $0$ of radius $\varepsilon$. To make the function be defined for $A = 0$, we can modify it as follows:

$$ s_4(A) = \frac{1}{\delta + \min_{x \in C \setminus B_{\varepsilon}(0)} \|Ax\|_2},$$

but, however, none of these functions is absolutely homogeneous.

So, my question is as follows: is it possible to construct the norm of matrix by the analogy with the $\|.\|_{C,2}$ norm but that uses the minimum instead of maximum?

Answer 1

Your $s_3(A)$ can be useful, since it is $\|A^{-1}\|^{-1}$ up to a constant, if $C$ is the closed unit ball. This is because, when $A^{-1}$ is invertible, $$c\|x\|\le\|Ax\|\iff c\|A^{-1}y\|\le\|y\|$$

If $C$ is separated from the origin (by a hyperplane), then all of the functions suffer from not being specific enough, i.e., $s(A)=0$ do not imply $A=0$. Since one of the main uses of norms is to define convergence, this defeats their purpose. This apart from not being homogeneous; for example, $s_4(0)=\frac{1}{\delta}$.