Let $(B_t)_{t\ge0}$ be a continuous (say wrt the Hausdorff metric) family of nonempty centrally symmetric convex compact subsets of $\Bbb R^2$. For a norm $\|\cdot\|$ on $\Bbb R^2$ and each real $t\ge0$, let $M_t$ denote the set of all maximizers $x$ of $\|x\|$ over $x\in B_t$.
Is it then always true that for each real $t\ge0$ there is some $x_t\in M_t$ such that the family $(x_t)_{t\ge0}$ is continuous?
It seems to me that it is not possible in general: consider e.g. $B_1$ to be the unit ball and $B_t$ for $t<1$ to be a family of ellipses tending to $B_1$ as $t$ approches 1 such that the semi-major axis are limiting to the segment joining the points $(1,0)$ and $(-1,0)$. Do the same from the right but for the semi-major axis limiting to the segments joining the points $(0,1)$ and $(0,-1)$. Then, it will not be possible to get continuity of the $x_t$ at the point $t=1$.
By the way, you can compare this problem with the Michael selection theorem (for lower semi-continuous maps with nonempty compact convex values).
