  A gravity theory without masses can be constructed in Minkowski spaces using
a geometric Minkowski potential. The related affine spacelike spheres can be
seen as the regions of the Minkowski spacelike vectors characterized by a
constant Minkowski gravitational potential. These spheres point out, for each
dimension $n \geq 3$, spacetime models, the de Sitter ones, which satisfy
Einstein's field equations in absence of matter. In other words, it is possible
to generate geometrically the cosmological constant. Even if a lot of possible
parameterizations have been proposed, each one highlighting some geometric and
physical properties of the de Sitter space, we present here a new natural
parameterization which reveals the intrinsic geometric nature of cosmological
constant relating it with the invariant affine radius coming from the so called
Minkowski-Tzitzeica surfaces theory.
