[Math] Cone and Dual Cone in $\mathbb{R}^2$ space

convex-analysisconvex-conedual-cone

Boyd's book, my understanding of cone and dual cone for 2-space is:
If we think of a circle in $\mathbb{R}^2$ space, cone $K$ and dual cone $K^*$ would be like this:

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Here, $K^* = y | x^Ty \geq 0 \text{ for all } x \in K$

Now question is: How do you draw cone and dual cone for these:

  1. $K = \{ (a_1, a_2) \in \mathbb{R} | |a_1| \leq a_2 \}$

  2. $ K = \{ Ax | x \geq 0 \}$

I am just trying to get an intuitive idea about the geometry of the cones and dual cones.

Best Answer

Your understanding is wrong. Consider the vector $x=(1,1)$ and the vector $y=(-1,-1)$. $x$ is clearly in $K$. You claim that $y$ is in $K^{*}$, but $x^{T}y=-2<0$. In fact, the dual of the positive orthant $R^{n}_{+}$ is $R^{n}_{+}$.

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