Séance Séminaire

Séminaire Gaston Darboux

Banach--Mazur compactum and Loewner ellipsoid.

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.