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Let $P$ be a simple $3$-dimensional (and full-dimensional) lattice polytope such that every facet $F$ is a smooth polytope. Is then $P$ itself smooth?

EDIT: A full-dimensional lattice polytope $P$ is called smooth if its normal fan $\Sigma_P$ is smooth.

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    $\begingroup$ What is a "smooth polytope"? $\endgroup$
    – Igor Rivin
    Commented Aug 14, 2018 at 18:02
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    $\begingroup$ Isn't the tetrahedron with vertices 000, 110, 101, 011 a counterexample? $\endgroup$
    – Richard Stanley
    Commented Aug 14, 2018 at 19:08
  • $\begingroup$ @RichardStanley It is, thanks a lot! $\endgroup$
    – Mellon
    Commented Aug 14, 2018 at 19:40

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