  We discussed some properties of a family of symmetric spaces, namely
$\mathcal{P}(n):=SL(n,\mathbb{R})/SO(n,\mathbb{R})$, where we replace the
Riemannian metric on $\mathcal{P}(n)$ with a premetric suggested by Selberg.
These properties are generalizations of properties on the hyperbolic spaces
$\mathbf{H}^n$ related to Poincar\'e's fundamental polyhedron theorem.
