Let $C$ be a (nonempty) convex compact subset of $\mathbb R^n$.

**General question:** Under what conditions on $C$ does the support function

$$\sigma_C(x) := \sup_{y \in C}x^Ty $$ define a norm on $\mathbb R^n$ ?

**Specific question:** For a real $n$-by-$n$ matrix $W = (w^j_i)$ , define $$C_W := \{y \in \mathbb R^n \text{ s.t } \sum_{i=1}^nw^j_i y_i^2 \le 1,\;\forall j\}.
$$
Under what conditions on $W$ does the $\sigma_{C_W}$
define a norm on $\mathbb R^n$ ?

**Observations:**

- In case $w^j_k \in \{0, 1\}$ and $\sum_{i=1}^n w^j_i > 0$ for all $k$ and $j$ (i.e $W$ is the adjacency matrix of a directed graph with no isolated nodes), then $\|.\|_W$ is indeed a norm called the
*overlapping group-norm*(see Lemmas 1 & 2 of this manuscript), where the groups are taken to be the adjacency lists of $W$, i.e $g_i := \{j | w^j_i = 1\}$. I'd like to extend this to a general (nonnegatively) weighted directed graph.