We study the luminosity function, the peculiar velocities, and the sizes of voids in the Local Volume (LV) in observational samples of galaxies which contain galaxies down to M_B = -10 and to M_B = -12 within the distance 4-8 Mpc. There are ∼30 voids with sizes ranging from 1 to 4.5 Mpc. We compare the observational distribution of sizes of voids with the voids in very high resolution simulations of the LCDM model with WMAP1 and WMAP3 parameters. The theoretical void function matches the observations remarkably well only if we use haloes with circular velocities V_c larger than 40-45 km/s (M_vir = (1-2)× 10¹⁰M) for models with with σ₈ = 0.9 and V_c > 35 km/s (M_vir = (6-8)×10⁹M) for σ₈ = 0.75. We exclude the possibility that in the LCDM model haloes with circular velocities < 35 km/s can host galaxies as bright as M_B = -12: there are too many small haloes in the LCDM model resulting in voids in the LV being too small as compared with the observations. The problem is that many of the observed galaxies with M_B = -12-14 have HI rotational velocities below 25 km/s that strictly contradicts the LCDM predictions. Thus, the LCDM model faces the same overabundance problem, which it had with the number of satellites in the LG: the theory predicts a factor of ten more haloes as compared with the observed number of dwarf galaxies. We also estimate the rms deviations from the Hubble flow H for galaxies at different distances from the Local Group and nd that in most of our model LV-candidates the rms peculiar velocities are consistent with observational values: σ_H = 50 km/s for distances less than 3 Mpc and σ_H = 80 km/s for distances less than 8 Mpc. At distances 4 (8) Mpc, the observed overdensities of galaxies are 3.5-5.5 (1.3-1.6) - signicantly larger than typically assumed.

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