Intersection of Convex Sets is Convex Set (Vector Spaces)

Theorem
Let $V$ be a vector space over $\R$ or $\C$.

Let $\left\{{L_i : i \in I}\right\}$ be a family of convex subsets of $V$, indexed by $I$.

Then the intersection $\displaystyle L = \bigcap_{i \mathop \in I} L_i$ is also a convex subset of $V$.

Proof
Let $x, y \in L$. Then by definition of set intersection, $\forall i \in I: x, y \in L_i$.

The convexity of the $L_i$ yields, for every $i \in I$:


 * $\forall t \in \left[{0 \,.\,.\, 1}\right]: tx + \left({1 - t}\right) y \in L_i$

Therefore, these elements are also in $L$, by definition of set intersection.

Hence $L$ is also convex.