  Monotone polytopes, also known as smooth reflexive polytopes, are the
polytopes associated to monotone symplectic toric manifolds and Gorenstein Fano
toric varieties. We first show that the only monotone polytopes admitting
blow-ups at vertices are the simplex and the result of a codimension-two
blow-up in it (this is the polyhedral version of a result of Bonavero from
2002). Then we show that the $n$-simplex admits disjoint blow-ups at faces if
and 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.
