Convex hull of one point in $\mathbb{R}^{2}$

Question

Let $S\in\mathbb{R}^{2}$ and let the size of $|S|=1$. If $x\in S$ then what is the convex hull of $S$? Is it $\{x\}$ or is it an empty set? Thank you.

Answer 1

By the definition of Convex Hull of a set S, it is the smallest convex set containing all points of S and since a singleton set is convex itself, thus convex hull of $\{x\}$ is $\{x\}$ itself.

Answer 2

If $S=\{x\}$, then the convex hull of $S$ is given by

$$\{tx+(1-t)x: t \in [0,1]\}.$$

We have $tx+(1-t)x=x$, hence the convex hull of $S$ is $S$.