# What is the difference between convex cone and convex hull?

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

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 $$X$$.
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.