Convex Set is Star Convex Set

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Theorem

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

Let $A \subseteq V$ be a non-empty convex set.


Then $A$ is a star convex set, and every point in $A$ is a star center.


Proof

Let $a \in A$.

Note that there is at least one point in $A$, as $A$ is non-empty.

If $x \in A$, then there is a line segment joining $a$ and $x$.

By definition of star convex set, it follows that $A$ is star convex, and $a$ is a star center.

$\blacksquare$