Core of a Convex Set is Convex

Let $X$ be a vector space and let $\Omega\subset X$ be convex, with $\operatorname{core}(\Omega)\neq\emptyset$.

Proposition: The set core(Ω) is convex.

Context: This is the algebraic analogue of the fact that the interior of a convex set (in a normed space) is convex; compare interior/closure of convex sets.