Singleton is Convex Set

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Theorem

Let $V$ be a vector space over $\R$ or $\C$, and let $v \in V$.


Then the singleton $S = \left\{{v}\right\}$ is a convex set.


Proof

For any $x, y \in S$, we have $x = y = v$.

It follows that:

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

Hence $S$ is a convex set.

$\blacksquare$


Sources