Definitions
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* Let $\mathbb{R}^n$ be the n dimensional Eucledean space.
* With $S \subseteq \mathbb{R}^n$, let $S^G$ be the set of all finite nonnegative linear combinations of elements of $S$.
* A set $K$ is defined to be a cone if $K = K^G$.
* A set is convex if it contains with any two of its points, the line segment between the points.

Questions
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1. From the above definitions, does it not follow that every cone is convex?
If not what will be an example of a nonconvex cone?
2. $K\_n = \left\{x \in \mathbb{R}^n : \sqrt{(x^2\_2 + \cdots + x^2\_n) } \le x\_1 \right\}$ is called ***the ice cream cone***. I can see how it is convex and the fact that it is a cone. But I do not understand the ice cream part.

Why is called the ice cream cone? how can i visuslise it or at least get some intuition on what it means.

It seems to be a set made up of infinite number of verctor dimensions i.e. $\{ n \in \mathbb{Z} : n \ge 2 \}$ and infinite vectors in each dimension. But I can't see the shape.

More generally, is the concept of a cone here in any way related to the geometric shape?