On compact hyperbolic Coxeter d-polytopes with d+4 facets
  We show that there is no compact hyperbolic Coxeter d-polytope with d+4 facets for d>7. This bound is sharp: examples of such polytopes up to dimension 7 were found by Bugaenko (1984). We also show that in dimension d=7 the polytope with 11 facets is unique. 
Anna Felikson, Pavel Tumarkin