How can the union $P\cup Q$ of two polytopes $P, Q$ also be a polytope?
About the union $P\cup Q$ of two polytopes $P, Q$
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polytopes
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0Why would it not be? – 2017-02-03
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1@HenningMakholm I suppose we should have $P\cap Q\neq \emptyset$ – 2017-02-03
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0So if $P\cap Q\neq \emptyset$ then $P\cup Q$ is a polytope? – 2017-02-03
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0The answer depends on the definition of a polytope. It's quite common to use the term "polytope" to denote *convex* polytopes. In that sense, the answer is in general no, even if their intersection is not empty. In the non-convex sense the answer is yes. Edit: Ups, I misread the question a little bit. In the convex case you have: $P\cup Q$ is a convex polytope if and only if $P \cup Q$ is equal to the convex hull of $P$ and $Q$. – 2017-02-07