Convex sets via convex combinations

Proposition. A nonempty set $\Omega\subset X$ in a real vector space $X$ is convex if and only if it contains every convex combination of finitely many of its points; i.e., for all $m\in\mathbb{N}$, all $x_1,\dots,x_m\in\Omega$, and all $\lambda_i\ge 0$ with $\sum_{i=1}^m\lambda_i=1$, we have

Context. This is the “finite averaging” characterization of convexity and is used to describe convex hulls and many convex constructions.

Proof sketch. The “only if” direction follows by induction on $m$ using the two-point definition of convexity. For “if,” take $m=2$ to recover the defining two-point condition.