# Dual polyhedron & dual cone

From Wiki:

1. Def. of dual of polyhedral (polytope): polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other.

EX:

2. Def. of dual cone: $$C^*=\{y \mid \langle y,x\rangle\geq 0, \forall x\in C\}$$.

Is there any relation between two definitions or can I obtain 1. from 2.?

Or is 1. closer to the definition of "polar of set $C$"?
$$C^o=\{y \mid \underset{x\in C}{\text {sup}} \big|\langle x,y\rangle \big|\leq 1\}$$.

The dual polyhedron corresponds to the dual set. In fact, for any set $$K \subset \mathbb R^n$$, you have
• $$K^\circ = (\operatorname{conv}(K \cup \{0\}))^\circ$$, where $$\operatorname{conv}$$ denotes the convex hull.
Now, let $$P = \operatorname{conv}(\{p_1, \ldots, p_N\})$$ be a polyhedron. Then, you have \begin{align}P^\circ &= \{p_1,\ldots, p_N\}^\circ = \bigcap_{i=1}^N \{p_i\}^\circ\\ &= \bigcap_{i=1}^N \{x \mid \langle x, p_i\rangle \le 1 \}.\end{align}
Thus, the points $$p_i$$ are the (potentially) extremal points in $$P$$, whereas they define the faces in $$P^\circ$$.
Finally, I would like to mention that the polar cone and the polar set are also closely related. Indeed, for $$K \subset \mathbb R^n$$, you can show that the polar cone of $$\{1\} \times K$$ is the closed conical hull of $$\{1\}$$ times the polar set of $$K$$. Conversely, $$\{1\}$$ times the polar set of $$K$$ is the intersection of $$\{1\} \times \mathbb R^n$$ with the polar cone of $$\{1\} \times K$$.