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


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.