  In this work we define a new pseudometric in $\mathcal K^n_*$, the hyperspace
of all non-degenerated compact convex sets of $\mathbb R^n$, which is invariant
under similarities. We will prove that the quotient space generated by this
pseudometric (which is the orbit space generated by the natural action of the
group of similarities on $\mathcal K^n_*$) is homeomorphic to the Banach-Mazur
compactum $BM(n)$, while $\mathcal K^n_*$ is homeomorphic to the topological
product $Q\times\mathbb R^{n+1}$, where $Q$ stands for the Hilbert cube.
Finally we will show some consequences in convex geometry, namely, we measure
how much two convex bodies differ (by means of our new pseudometric) in terms
of some classical functionals.

Metric Geometry;Geometric Topology