Intersection of Convex Sets is Convex Set (Vector Spaces)

Theorem

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

Let $\CC$ be a family of convex subsets of $V$.

Then the intersection $\ds \bigcap \CC$ is also a convex subset of $V$.

Proof

Let $x, y \in \ds \bigcap \CC$.

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

The convexity of each $C$ yields:

$\forall t \in \closedint 0 1: t x + \paren {1 - t} y \in C$

Therefore, these elements are also in $\ds \bigcap \CC$, by definition of set intersection.

Hence $\ds \bigcap \CC$ is also convex.

$\blacksquare$

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 2.$ Orthogonality