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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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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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    @P23 Yes, it is enough to t$a$ke 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, $b$ut also the convex hull of the finitely many vertices.2011-10-30