Page 17 of this document states that, "it is not true, in general, that the sum of two closed, covex sets is closed."
Question. Is there a straightforward example of this phenomenon?
According to the linked document, both summands have to be unbounded before this can occur.
Consider $A=\{(x,y)\,:\, xy\ge 1,\ x\ge 0\}$ and $B=\{(-x,y)\,:\, (x,y)\in A\}$.
Then, $A+B=\Bbb R\times (0,\infty)$: if $B$ slides horizontally along the half-line $[x,\infty)\times\left\{\frac1x\right\}$, the whole upper half plane $\{y> \frac1x\}$ gets covered; doing it for all $x\ge0$ covers $\{y>0\}$ and, of course, $A+B$ cannot contain points $(\alpha,\beta)$ with $\beta\le 0$.