Linear Closure of a Convex Set is Convex

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

Proposition: The linear closure $\operatorname{lin}(\Omega)$ is convex.

Context: The definition of $\operatorname{lin}(\Omega)$ is built from line segments . Convexity of $\Omega$ ensures that “segments staying in $\Omega$” is stable under convex combinations , which yields convexity of $\operatorname{lin}(\Omega)$.