What is the difference between convex cone and convex hull?

Question

I am reading this definition of convex cone and this definition of the convex hull of a finite set of points and I am in trouble in understanding the difference.

Am I right that, given a set of points $x_1,..., x_n$ in $R^m$, convex cone is defined by

$$y \in R^m | y=\sum a_i x_i, a_i \geq 0, i=1,...,n $$

and convex hull same than above but with

$\sum a_i=1$ in supplement ?

Thanks for clarification. A "geometric" example on a 2D dataset would be appreciated

Answer 1

The two are very different in definition. Definitions:

A set $S\subseteq \mathbb R^n$ is a convex cone if, for any $x\in S$ and any positive real $\alpha$, the vector $\alpha x$ is also an element of $S$. That is, $S$ is a convex cone if and only if $$\forall x\in S\forall \alpha\in[0,\infty):\alpha x\in S$$

For a set $X$, the convex hull of $X$ is the smallest convex set that contains $X$.

You can see a very big difference in the two definitions in the fact that a convex hull is defined in terms of another set, while "convex cone" is a property that a set can either have or not.

That is, it is possible to take a set, $S$, and ask "is $S$ a convex cone?". This question has a yes or no question, depending on $S$. On the other hand, the question "is $S$ a convex hull" doesn't have a yes or no question. You have to change the question to "Is $S$ a convex hull of $X$" before you can answer it.