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Along with a polytope P one has the notion of its dual which is officially defined via the inner product. However, in three dimensions at least, the dual is often pictured simply by placing a point in each face of P and then taking the convex hull. Will this same method work in general?

Question: Let P be an n-dimensional polytope. Place points at the barycenter of each facet of P and designate by $\,$Q$\,$ the convex hull of these points. $\,$Is the resulting polytope$\,$ Q$\,$ combinatorially equivalent to the dual of$\,$ P$\,$?

Thanks.

user2052
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  • This works almost never, not even in 3D (or 2D). This technique works only for the most symmetric cases, and requires at least that the polytope has an insphere, but this is far from sufficient. – M. Winter Sep 26 '20 at 23:22

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No. For example, let P be a regular icosahedron with each vertex perturbed randomly by a small amount. Your constructed Q will consist of 36 triangles, rather than 12 pentagons.

Don Hatch
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