Determine the dual cone $\mathbb{K}^{*}$

Question

In the convex optimization book, dual cone of a cone $K$ is:

$$ \mathbb{K}^{*} = \left \{ y^Tx \ge 0 \textrm{ for all } x \in K\right \} $$ What is dual cone $\mathbb{K}^{*}$ of the follow cone $\mathbb{K} = \mathbb{R}_{+}^{n}$ ?

Answer 1

$\mathbb K^* = \mathbb K$. Indeed, on one hand $x\in \mathbb K$ for all $y\in \mathbb K$, $\left\langle x, y\right\rangle = \sum_{i=1}^n x_iy_i \ge 0.$ and on the other hand, if $x\in \mathbb K^*$ then $x_i = \left\langle x, \underbrace{e_i}_{\in \mathbb K}\right\rangle\ge 0$ so $x\in \mathbb K$.