1

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.

bencemeszaros
  • 183
  • 1
    Key point : in the definition of the convex hull which is probably the union of all line segments between all points of the set, this includes lines between a point and the same point i.e. the point itself. So even though you are thinking " there are no line segments here", there is one, of length $0$, namely the point $x$ itself. Therefore, the convex hull of the singleton ${x}$, is itself. In other words, it is convex. – Sarvesh Ravichandran Iyer Mar 07 '18 at 12:11
  • Yeah the notation of the convex set definition what confused me.. Please see my answer to the top comment. – bencemeszaros Mar 07 '18 at 12:30

2 Answers2

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 $S$.

Fred
  • 77,394
2

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.

  • "Def. A subset $C$ of $\mathbb{R}^{n}$ is said to be convex if $(1-\lambda)x+\lambda y\in C$ whenever $x\in C$, $y\in C$ and $0<\lambda<1.$" So it is not followed by the notation of this definition then that $x$ and $y$ is not the same? Would it have to state $x\neq y$ explicitly? – bencemeszaros Mar 07 '18 at 12:17
  • @bencemeszaros Actually the definition is $\forall x,y \in C$ . And for dealing with your confusion, observe that $\lambda$ is chosen to be $0 < \lambda <1$ not $0 \le \lambda \le 1$ . Does that ring a bell? –  Mar 07 '18 at 12:32
  • It does not follow that $x \neq y$! In particular, $x = y$ is part of the proposition. Therefore, every set is contained in its convex hull. Note that there is no explicit mention of $x \neq y$. It is only written $x,y \in C$. Line segments can go from one point to the same point, then they'd be of length $0$ and consist of the point alone. – Sarvesh Ravichandran Iyer Mar 07 '18 at 12:33