  This article commences a study of convergence of discretized point-object
configurations, which we call Brill-Lindquist-Riemann sums, towards a charged
dust continuum from the perspective of relativistic initial data. We are
motivated by the interpretation of Brill-Lindquist-Riemann sums as collections
of relatively isolated astrophysical bodies such as stars and galaxies in the
universe, and the interpretation of the dust continuum as the universe itself.
Our work begins by establishing the existence and the uniqueness of
horizons/minimal surfaces of Brill-Lindquist metrics in the vicinity of the
point-sources ("stars"). We then study the geometries of the regions exterior
to said minimal surfaces, and discuss their Gromov-Hausdorff and intrinsic flat
limit. An interesting and purely geometric byproduct of our work are examples
in which the scalar curvature jumps upon taking Gromov-Hausdorff and /or
intrinsic flat limits.
