Here's the question, as in the textbook (Real Mathematical Analysis, Pugh).
The unit ball with respect to a norm $||\, \cdot \,||$ on $\mathbb{R}^2$ is $$ \{ v \in \mathbb{R}^2 : ||\, v \,|| \leq 1 \} . $$ a. Find necessary and sufficient geometric conditions on a subset of $\mathbb{R}^2$ that it be the unit ball for some norm.
b. Find necessary and sufficient geometric conditions that a subset of $\mathbb{R}^2$ be the unit ball for a norm arising from an inner product.
I have proven a. by specifying for a set $V$ the conditions of: convexity of $V$, existence of $\max \{ a \in \mathbb{R} : av \in V \}$ for all non-zero $v \in V$, and $0 \in V$. The 'necessary' direction is rather straightforward, while the 'sufficient' direction can be shown using construction and defining the norm by the reciprocal of the above maximum value. Note: I haven't done any (general) topology yet, though I've been told that the existence of the maximum of the aforementioned set is equivalent to closedness and boundedness (compactness).
As for (b), my intuition says that the set must be an ellipse (not sure if axis-aligned or generally), but I really don't know where to begin in proving this. Any advice would be appreciated.