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Let $S \subseteq \mathbb{R}^d$ be a $d$-dimensional convex set (i.e. $\exists d+1$ affinely independent points in $S$). Let the origin of the coordinate system lie in the interior of $S$ and let: $$S^* = \underset{a\in S}\bigcap K(a) $$ where $K(a) = a \cdot x \le 1$

Then, $S^*$ is the polar of $S$. Also, if $S$ is a $d$-polytope in $\mathbb{R}^d$, then $S^*$ is a dual of $S$.

Since a $d$-polytope can be represented by a countably infinite set of points, this implies that the dual (which is also a polytope) is formed from the intersection of a countably infinite number of halfspaces --- which clashes with the definition of a polytope as the intersection of finitely many closed halfspaces.

What am I doing wrong here?

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up vote 1 down vote accepted

If the polytope can be represented by a countably infinite set of points, the dual can be represented by the intersection of countably infinite number of halfspaces.

Which is true and not in contradiction with the existence of a finite representation.

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OK, I see. So that means there exists a representation of the dual with finitely many halfspaces as well? –  Jacob Oct 30 '11 at 18:29
    
@P23 Yes, it is enough to take the halfspaces corresponding to the vertices of the original polytope. This is analogous to the fact that the original polytope is the convex hull of all its points, but also the convex hull of the finitely many vertices. –  Phira Oct 30 '11 at 18:50
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