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

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?

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, $$s_4(0)=\frac{1}{\delta}$$.