  Introduced by Kodama and Williams, Bruhat interval polytopes are generalized
permutohedra closely connected to the study of torus orbit closures and total
positivity in Schubert varieties. We show that the 1-skeleton posets of these
polytopes are lattices and classify when the polytopes are simple, thereby
resolving open problems and conjectures of Fraser, of Lee--Masuda, and of
Lee--Masuda--Park. In particular, we classify when generic torus orbit closures
in Schubert varieties are smooth.
