Consider a norm $\|\ \|$ on $\mathbb{R}^2$ s.t. $\|\ \|$ is smooth, strict convexity, i.e. $$\|v\|=\|w\|=1,\ v\neq w \Rightarrow \bigg\| \frac{v+w}{2}\bigg\|<1$$ Define a function $$f: D\times D\rightarrow \mathbb{R},\ f(v,w)=|{\rm det}\ [v\ w]|$$ where $D$ is $\|\ \|$-unit ball.
Problem : When $f$ has a maximum at $(v_0,w_0)$, then convex hull of $\{v_0\pm w_0, -v_0\pm w_0 \}$ contains $D$.
(Here I can not convince that we need the condition where $\partial D$ is smooth.)
Remark : When $P$ is a parallelogram circumscribing $\|\ \|$-unit circle, then ${\rm area\ conv}\ (P)$ is minimum when $P$ is realized under the above way.