A bounded convex domain $\Omega$ in $R^n$ admits a projectively invariant
Riemannian metric coming from a special hypersurface (the hyperbolic
affine sphere) asymptotic to the cone $\mathcal{C}$ over $\Omega$ in
$R^{n+1}$. We will show an explicit relationship between this affine
sphere and the complete K\"ahler-Einstein metric on the tube domain over
$\mathcal{C}$. We will also discuss how to relate this to the
differential geometry and PDEs native to manifolds with restricted
coordinate charts (Thurston's $(X,G)$ manifolds), in particular affine
flat and projectively flat manifolds.