Segments from Core Points Stay in the Core

If a is in core(Ω) and b in Ω, then points on [a,b) remain in core(Ω).

Segments from Core Points Stay in the Core

Let $X$ be a vector space and let $\Omega\subset X$ be convex .

Proposition: If $a\in\operatorname{core}(\Omega)$ (see core ) and $b\in\Omega$ , then

[a,b)\subset \operatorname{core}(\Omega),

where $[a,b)$ is the half-open line segment from $a$ to $b$ .

Context: This is the “core” analogue of the interior segment lemma for normed spaces. It is used to prove structural properties of $\operatorname{core}(\Omega)$ such as idempotence.