Convex Set

Last updated Nov 1, 2022

Let $V$ be a Vector Space over $\mathbb{R}$ and $S \subset V$. $S$ is a Convex Set if for all $\lambda \in [0,1]$ and all $u, v \in S$ $$\lambda u + (1 - \lambda) v \in S$$

1. $S$ is a Convex Set If and Only If $S$ contains all Closed Line Segments between its points. This is just a restatement of the definition of a Convex Set.
2. A Set is Convex iff it contains all of its Convex Combinations

• Real Numbers