  CONTEXT: Cosmic voids are observed in the distribution of galaxies and, to
some extent, in the dark matter distribution. If these distributions have
fractal geometry, it must be reflected in the geometry of voids; in particular,
we expect scaling sizes of voids. However, this scaling is not well
demonstrated in galaxy surveys yet. AIMS: Our objective is to understand the
geometry of cosmic voids in relation to a fractal structure of matter. We
intend to distinguish monofractal voids from multifractal voids, regarding
their scaling properties. We plan to analyse voids in the distributions of mass
concentrations (halos) in a multifractal and their relation to galaxy voids.
METHODS: We make a statistical analysis of point distributions based on the
void probability function and correlation functions. We assume that voids are
spherical and devise a simple spherical void finder. For continuous mass
distributions, we employ the methods of fractal geometry. We confirm the
analytical predictions with numerical simulations. Smoothed mass distributions
are suitable for the method of excursion sets. RESULTS: Voids are very
nonlinear and non-perturbative structures. Voids reflect the fractal geometry
of the matter distribution but not always directly: scaling sizes of voids
imply fractal geometry, but fractal voids may have a complicated geometry and
may not have scaling sizes. Proper multifractal voids are of this type. A
natural multifractal biasing model implies that the voids in the galaxy
distribution inherit the same complicated geometry. CONCLUSIONS: Current galaxy
surveys as well as cosmological N-body simulations indicate that cosmic voids
are proper multifractal voids. This implies the presence in the voids of
galaxies or, at least, small dark matter halos.
