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

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

• Your question is very unclear. Please add more explanation and correctly format the math characters using Latex. Feb 9, 2017 at 4:17
• iam not able to find much information Mr joshua . thats why iam asking this question :\ . Feb 9, 2017 at 4:24
• Why is no answer accepted? Jul 24, 2020 at 2:44

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

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

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:

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