Intersection of Convex Sets is Convex

Last updated Nov 1, 2022

Let $V$ be a Vector Space on $\mathbb{R}$ and let $\mathcal{S} \subset \mathcal{P}(V)$ be a collection of Convex Sets. Then $\bigcap\limits_{S \in \mathcal{S}}S$ is a Convex Set.

Let $T = \bigcap\limits_{S \in \mathcal{S}}S$ and let $u, v \in T$. Suppose $\lambda \in [0,1]$. Then $\forall S \in \mathcal{S}$, we have that $\lambda u + (1 - \lambda) v \in S$. Thus $\lambda u + (1 - \lambda) v \in T$ and $T$ is a Convex Set. $\blacksquare$

Intersection of Structures is still a Structure