By Banach--Mazur compactum $Q(n)$ it is understood the space of
isometry classes of all $n$-dimensional Banach spaces with
Banach--Mazur distance.
This space has the representation $Conv (R^n)/GL(n,R)$,
where $Conv (R^n)$ means the space of all convex symmetric
bodies in $R^n$ with Hausdorff distance.
There was a problem in geometric functional analysis to
study the topology of Banach--Mazur compactum.
Ageev, ... proved that $Q(n)$ is an absolute retract,
$Q(2)$ is not homeomorphic to Hilbert cube $I^\infty $,
but the complement in $Q(n)$ to the point corresponding to Eucledean space
is an $I^\infty $-manifold.
The proofs are based on the usage of Loewner ellipsoid.