  We show that in any Euclidean space, an arbitrary probability measure can be
reconstructed explicitly by its geometric (or spatial) distribution function.
The reconstruction takes the form of a (potentially fractional) linear PDE,
where the differential operator is given in closed form. This result implies
that, contrary to a common belief in the statistical depth community, geometric
cdf's in principle provide exact control over the probability content of all
depth regions. We present a comprehensive study of the regularity of the
geometric cdf, and show that a continuous density in general does not give rise
to a geometric cdf with enough regularity to reconstruct the density pointwise.
Surprisingly, we prove that the reconstruction displays different behaviours in
odd and even dimension: it is local in odd dimension and completely nonlocal in
even dimension. We investigate this issue and provide a partial counterpart for
even dimensions, and establish a general representation formula of the
geometric cdf of spherically symmetric probability laws in odd dimensions. We
provide explicit examples of the reconstruction of a density from its geometric
cdf in dimension 2 and 3.
