Let $A\in\mathbb C^{n\times n}$. If $\|\cdot\|$ is a norm on $\mathbb C^n$, then $\|A\| = \sup_{\|x\|=1}\|Ax\|$. I will call such a matrix norm a vector matrix norm because it is defined by means of a vector norm. The following is well known:
The infimum of $\|A\|$ over all vector-matrix norms equals the spectral radius of $A$.
Now, given a scalar product $(\cdot,\cdot)$ on $\mathbb C^n$, the numerical range of $A$ is given by $$ W_{(\cdot,\cdot)}(A) = \{(Ax,x) : \|x\|=1\}, $$ where $\|\cdot\|$ is the vector norm induced by the scalar product $(\cdot,\cdot)$. It is also well known that the spectrum $\sigma(A)$ of $A$ (set of eigenvalues) is contained in $W(A)$ for any scalar product and that $W(A)$ is convex. Now, here goes the question:
Denoting by $\mathcal S$ the set of scalar products on $\mathbb C^n$ and by $\operatorname{conv}$ the convex hull, is it true that $$ \bigcap_{(\cdot,\cdot)\in\mathcal S}W_{(\cdot,\cdot)}(A) = \operatorname{conv}(\sigma(A))\,?$$
After some research on the web, I could find an article that actually answers my question in the affirmative:
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.590.4912&rep=rep1&type=pdf
and there Thm. 6.3. It's based on the following paper:
S. Hildebrandt, ̈Über den Numerischen Wertebereich eines Operators, Math. Ann. 163 (1966), 230–247.