How can the union $P\cup Q$ of two polytopes $P, Q$ also be a polytope?
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Why would it not be? – hmakholm left over Monica Feb 03 '17 at 08:07
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1@HenningMakholm I suppose we should have $P\cap Q\neq \emptyset$ – Ewan Delanoy Feb 03 '17 at 08:11
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So if $P\cap Q\neq \emptyset$ then $P\cup Q$ is a polytope? – X. Gao Feb 03 '17 at 08:15
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The 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 $Q$. – Willem Hagemann Feb 07 '17 at 12:57