*Note: This is very similar to this question of mine on Math.Se, which hasn't received any answers yet.*

It is known, that for any centrally symmetric convex polygon $A$ there exists a norm $\| \cdot \|$, whose unit ball is $A$, namely $\| x \| = \inf \{ t>0: \frac{x}{t} \in A\}$.
But how about the other way around, given a "composite" norm
$$
\| \cdot \|
:= \sum_{k \in \mathbb{N}} \alpha_k \| \cdot \|_k,
$$
on a vector space $V$, where $\sum_{k \in \mathbb{N}} \alpha_k = 1$, such that the norm converges and the $\{i_k\}_{k \in \mathbb{N}}$ just being some index set for the norms used, **what can we say about the unit ball centred at zero with respect to $\| \cdot \|$, $\mathscr{B} := \{ \omega \in V: \| \| \omega \| < 1 \}$**.

It is clear that that a a polygon with a odd number of vertices can not occur because of the symmetry of the norm.