  Using high resolution N-body simulations we address the problem of emptiness
of giant 20 Mpc/h diameter voids found in the distribution of bright galaxies.
Are the voids filled by dwarf galaxies? Do cosmological models predict too many
small dark matter haloes inside the voids? Can the problems of cosmological
models on small scales be addressed by studying the abundance of dwarf galaxies
inside voids? We find that voids in the distribution of 10^12 Msun/h haloes
(expected galactic magnitudes ~ M_*) are almost the same as the voids in 10^11
Msun/h haloes. Yet, much smaller haloes with masses 10^9 Msun/h and circular
velocities v_circ about 20 km/s readily fill the voids: there should be almost
1000 of these haloes in a 20 Mpc/h void. A typical void of diameter 20 Mpc/h
contains about 50 haloes with v_circ > 50 km/s. The haloes are arranged in a
pattern, which looks like a miniature Universe: it has the same structural
elements as the large-scale structure of the galactic distribution of the
Universe. There are filaments and voids; larger haloes are at the intersections
of filaments. The only difference is that all masses are four orders of
magnitude smaller. There is severe (anti)bias in the distribution of haloes,
which depends on halo mass and on the distance from the centre of the void.
Large haloes are more antibiased and have a tendency to form close to void
boundaries. The mass function of haloes in voids is different from the
``normal'' mass function. It is much steeper for high masses resulting in very
few M33-type galaxies (v_circ about 100 km/s). We present an analytical
approximation for the mass function of haloes in voids.
