Examples of a complement of a convex set being convex

Question

I am having trouble thinking of any examples of a complement of a convex set being convex. It just doesn't make any sense to me as to how the complement can be convex instead of concave. The only one I can even think of would be the complement of the empty set, but is the empty set even convex?

Answer 1

A convex set is one in wich for every two points in the set, the line connecting them also lies $\textbf{entirely}$ within the set.

For an exemple of a convex set whose complement is also convex just consider the upper half-plane, i.e take $\mathbb{R}^2$ with standard coordinates $x,y$ and consider the set $$U=\{(x,y)\in\mathbb{R}^2 : y>0\}$$ it is clearly convex and so is its complement.