The Banach-Mazur compactum is the Alexandroff compactification of a   Hilbert cube manifold
  We prove that for every $n>2$, the Banach-Mazur compactum Q(n) is the compactification of a Hilbert cube manifold by the Euclidean point. For $n=2$ this result was proved earlier. 
Sergei M. Ageev, Semeon A. Bogatyi and Du\v{s}an Repov\v{s}