## Mark Broadie

*A theorem about antiprisms*

**Abstract:**

Let *P* be a polytope in **R**^{n} containing the origin in its interior, and let *P** be the algebraic dual polytope of *P.* Let Q **R**^{n} x [0,1] be the (*n*+1)-dimensional polytope that is the convex hull of P x {1} and *P** x {0}. For each face *F* of *P*, let Q(*F*) denote the convex hull of *F x {1} and **F** x {0}, where *F** is the dual face of *P*.* Then *Q* is an *antiprism* if the set of facets of *Q* is precisely the collection {*Q (F)*} for all faces *F* of *P.* If *Q* is an antiprism, the correspondence between primal and dual faces of *P* and *P** is manifested in the facets of *Q.* In this paper, necessary and sufficient conditions for the existence of antiprisms are stated and proved.

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