Can I write $\mathbb{R}^n_+$ as a norm cone?

Question

Let $\mathbb{R}^n_+=\{x=(x_1,\dots,x_n):x_i\geq 0,\forall i \},n\geq 2$. I wonder whether I can write $\mathbb{R}^n_+$ as a norm cone, i.e., $$\exists A, c, \|\cdot\|, s.t. x \in \mathbb{R}^n_+ \iff \|Ax\|\leq c^Tx.$$

I guess not but I could not figure out why. I have tried to look at the dual cone but it seems very hard to compute the dual cone of $\{x:\|Ax\|\leq c^Tx\}$.

Answer 1

It seems that you can. Take $\|\cdot\|$ to be the $1$-norm, $A = I$, and $c = (1,\dots,1)^T$. In particular, we have $$ x \in \Bbb R^n_+ \iff |x_1| + \cdots + |x_n| \leq x_1 + \cdots + x_n $$