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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:

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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 $C$"?
$$C^o=\{y \mid \underset{x\in C}{\text {sup}} \big|\langle x,y\rangle \big|\leq 1\}$$.

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1 Answer 1

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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 $\{1\} \times K$.

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