Intersection of Convex Sets is Convex Set (Vector Spaces)

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Theorem

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

Let $\mathcal C$ be a family of convex subsets of $V$.


Then the intersection $\displaystyle \bigcap \mathcal C$ is also a convex subset of $V$.


Proof

Let $x, y \in \displaystyle \bigcap \mathcal C$.

Then by definition of set intersection, $\forall C \in \mathcal C: x, y \in C$.

The convexity of each $C$ yields:

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

Therefore, these elements are also in $\displaystyle \bigcap \mathcal C$, by definition of set intersection.

Hence $\displaystyle \bigcap \mathcal C$ is also convex.

$\blacksquare$


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