Can a conic-hull be not trivial and bounded?

Question

Consider $X\subseteq\mathbb{R}^n$

The conic-hull of $X$ is cone$(X)=\{\lambda_1x_1+...+\lambda_nx_n\ :\ \lambda_i\geq0\ ,\ x_i\in X\}$

Can that set ever be not just $\{0\}$ and be bounded?

Answer 1

Suppose $\text{cone}(X)\ne{0}$.

Since $\text{cone}(\{0\})=\{0\}$, it follows that $X \ne \{0\}$.

Let $x\in X$, with $x\ne 0$.

Then, by definition of $\text{cone}(X)$, we have $\lambda x \in \text{cone}(X)$, for all $\lambda \ge 0$.

But $\{\lambda x\mid \lambda \ge 0\}$ is unbounded.

Therefore $\text{cone}(X)$ is unbounded.