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