What is a cone in $\mathbb R^n$?

Question

What is a cone in $\mathbb R^n$ ? By logical, I would give the definition :

$$\exists !p\in C: \forall x\in C, [x,p]\subset C,$$ or equivalently $$\exists !p\in C: \forall x\in C, \forall t\in [0,1], p+tx\in C.$$

Could such a definition work ? Is there an other definition ?

score 1 · Answer 1 · answered Jul 22 '18 at 22:20

1

A set $K\subseteq\mathbb{R}^n$ is a cone if for each $x\in K \text{ and }\alpha\in~\mathbb{R}_{++}$ we have $\alpha x\in K$. That is, $K$ is a cone if $\alpha K\subset K$ for each $\alpha\in\mathbb{R}_{++}$.

I would translate this definition to a more logical vocabulary as:

$$\forall \alpha\in\mathbb{R}_{++} (x\in K \implies \alpha x\in K).$$

answered Jul 22 '18 at 22:20

Ariel Serranoni

494

I don't know what is $\mathbb R _ {++}$. – user330587 Jul 22 '18 at 22:22
@user330587 $\Bbb R_{++}$ usually denote $(0,\infty)={x\in\Bbb R\mid x>0}$ – ℋolo Jul 22 '18 at 22:27
This is the same as the definition in the OP, with the unique $p$ chosen to be $0$. – preferred_anon Jul 22 '18 at 22:29
The problem with your definition it's that $\mathbb R^2$ would be a cone in $\mathbb R^3$, and so it's a little bit to general, no ? – user330587 Jul 22 '18 at 22:36
@Holo exactly! And user330587, if you mean something like $(x,y,0)$ is a cone in $\mathbb{R}^3$ you are right and there is no problem with this. – Ariel Serranoni Jul 23 '18 at 05:10
So a set of half-lines that almost touch the origin are a cone? I didn't know that. My geometric intuition seems to fit only a convex cone. – lightxbulb Oct 25 '23 at 01:25

What is a cone in $\mathbb R^n$?

1 Answers1