Segments from interior points stay in the interior

Lemma. Let $X$ be a normed vector space and let $\Omega\subset X$ be a convex set with nonempty interior. If $a\in \mathrm{int}(\Omega)$ and $b\in\Omega$, then

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

Context. This is a key geometric fact for convex sets: interior points “see” the whole set through interior segments. It underlies closure/interior relations for convex sets.

Proof idea. Starting from a ball around $a$ contained in $\Omega$, use convexity and scaling properties of balls to build a ball around each point $\lambda a+(1-\lambda)b$ (with $\lambda\in(0,1]$) that still lies in $\Omega$.