Conditions for the support function of ellipsoid to define a norm

Question

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.

Answer 1

General question: If $C$ is a subset of $\Bbb R^n$, then $\sigma_C$ is a norm if and only if $C$ is compact, convex, symmetric about the origin and has non-empty interior.

Proof: We know that $\|\cdot\|$ is a norm on $\Bbb R^n$ if and only if there exists a set $C$ with the properties listed above and such that $\|x\|=\gamma_C(x)$ where $\gamma_C(x)=\inf\{t>0\colon x/t \in C\}$. Note that $C$ is the unit ball of $\|\cdot\|$. Now, the dual norm of $\|\cdot\|$ is given by $\|x\|_*=\sigma_C(x)$. In particular, $\gamma_C$ is a norm if and only if $\sigma_C$ is a norm. More details can be found here and in the book on convex optimization of Boyd and Vandenberghe.

Specific question: $\sigma_{C_W}$ is a norm if and only if for every $i$, there exists $j$ such that $w_i^j>0$.

Proof: Note that if $\|\cdot\|_p$ denotes the usual $p$-norm, then $C_W$ is the unit ball of $$\|x\| = \big\|\|x\|_{w^1,2},\ldots,\|x\|_{w^n,2}\big\|_{\infty},$$ where $\|y\|_{w^j,2}$ is the weighted $2$-norm with (nonnegative) weights $w_1^j,\ldots,w_{n}^j$. From the previous observation, we know that $\sigma_{C_W}$ is a norm if and only if $\|\cdot\|$ is a norm. Clearly, $\|\cdot\|$ is nonnegative, (positively) homogeneous and satisfies the triangle inequality. Finally, you need to ensure that $\|x\|=0$ if and only if $x=0$. One direction is obvious. For the other direction, note that if $\|\cdot\|$ is a norm, then $\|e_i\|>0$ for every $i$, where $e_1,\ldots,e_n$ is the canonical basis of $\Bbb R^n$ so that for every $i$, there exists $j$ such that $0<\|e_i\|_{w^j,2}^2=w^j_i$.