Understanding Cones in general and the Ice cream Cone 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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*From the above definitions, does it not follow that every cone is convex?
If not what will be an example of a nonconvex cone?

*$K_n = \left\{x \in \mathbb{R}^n : \sqrt{(x^2_2 + \cdots + x^2_n) } \le x_1 \right\}$ 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?
 A: "Ice cream" is not a commonly used technical term in this context, so don't take it too seriously. It looks like it's just one author's attempt at a cutesey and memorable name.
A more dignified term for this shape in the $n=3$ case would be (the interior of one sheet of) a right-angled circular cone.
Whether it has anything to do with ice cream is a cultural matter. Where I come from, ice cream is often served in cones rolled from sweet flatbread with a top angle somewhere between 20° and 45°; they don't really match your shape. But I think that at least some places in the English-speaking world cones for ice cream are indeed right-angled (and made from cast sugar without flour).
A: Of course a cone is convex. Let $x^1,x^2 \in S^G$. Then $x^i = \sum_{s \in S}\xi^i_s s$, where all $\xi^i_s \ge 0$ and only finitely many $\xi^i_s \ne 0$. For each $t \in [0,1]$ we have $\eta_s = t \xi^1_s + (1 - t) \xi^2_s \ge 0$ and $\eta_s \ne$ only for finitely many $s$. Hence $t x^1 + (1-t)x^2 = \sum_{s \in S}\eta_s s \in S^G$.
The name "ice-cream cone" has been explained in Michal Adamaszek's comment.
