Understanding Cones in general and the Ice cream Cone

Question

Definitions

• 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

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?

Answer 1

"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).

Answer 2

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.