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Monotone polytopes, also known as smooth reflexive polytopes, are thepolytopes associated to monotone symplectic toric manifolds and Gorenstein Fanotoric varieties. We first show that the only monotone polytopes admittingblow-ups at vertices are the simplex and the result of a codimension-twoblow-up in it (this is the polyhedral version of a result of Bonavero from2002). Then we show that the $n$-simplex admits disjoint blow-ups at faces ifand only if the faces are disjoint and have dimensions adding up to $n-1$ or$n-2$. These results answer a question posed by Dusa McDuff in 2011.

The structure of monotone blow-ups in symplectic toric geometry and a question of McDuff