Affine Sets are Convex

Last updated Nov 1, 2022

Let $V$ be a Vector Space over $\mathbb{R}$ and $S \subset V$ be an Affine Set. Then $S$ is a Convex Set.

Let $u, v \in S$ and let $\lambda \in [0,1]$. Since $\lambda \in \mathbb{R}$ and $S$ is affine, we have that $$\lambda u + (1 - \lambda) v \in S$$ so $S$ is convex.