Subsets of a vector space that are convex and have convex complement.

Question

Let $V$ be a vector space over some field $F$.

Is there a characterization of the subsets $A$ that are convex and also have convex complement? Maybe when the dimension is finite it is possible?

I think that in $\mathbb R^n$ we can define it recursively, by using nested semihyperplanes.

Answer 1

Let $A \subseteq R^n$ be a convex nonempty subset of $R^n$ whose complement is convex as well. Since $A \cap A^c = \emptyset$ by separation theorem , we can separate them by a nonzero hyperplane. Therefore $A$ covers completely one side of the hyperplane and $A^c$ does same job in the other side of hyperplane. The contribution of $A$ and $A^c$ on the hyperplane is same as question in one less dimension. Therefore to realize the characterization of such sets you need think recursively.

For examples in $R^2$ there are three types of a such this set, 1- $A= R^2$ , 2- $A$ be closed half space,

3- $A$ be open half spaces $\cup$ half line

I think in $R^n$ we have $n+1$ different types of such set.