What Is a Pointed Cone Intuitively? How Could One Visualize It?

Question

A cone $K$, where $K ⊆\Bbb R^n$ , is pointed; which means that it contains no line (or equivalently, $(x ∈ K~\land~ −x∈K) ~\to~ x=\vec 0$.

Answer 1

Here is a picture (in 3D) of a cone which is not a pointed cone:

https://upload.wikimedia.org/wikipedia/commons/thumb/7/72/DoubleCone.png/1024px-DoubleCone.png

Here is a picture (in 3D) of a cone which is a pointed cone:

https://upload.wikimedia.org/wikipedia/commons/e/e7/Circular-pyramid.png

Answer 2

It means there are no 2 points inside it which creates a line and the whole line is contained by the cone.

For instance, take $ \mathbb{R}^{2} $ it is clearly a cone yet it is not pointed as any line in $ \mathbb{R}^{2} $ is contained by $ \mathbb{R}^{2} $.

Yet if you take $ \mathbb{R}^{2}_{++} $, namely only the right up quarter of it (Where each coordinate is non negative) it is a cone clearly, moreover it is a pointed cone as there is no line contained in it.

Remember that a line is defined by all points which are defined by $ {x}_{1}, {x}_{2} $ and $ \theta \in \mathbb{R} $ in the following way:

$$ \theta {x}_{1} + \left( 1 - \theta \right) {x}_{2} $$