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 .
When we compare the results with the predictions of the standard cosmological LCDM model , we find that the theory faces a sever problem : it predicts a factor of ten more dwarf haloes as compared with the observed number of dwarf galaxies .
In the LV we identify voids with sizes ranging from 1 to 4.5 Mpc and compare the observational distribution of void sizes 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 ) \times 10 ^ { 10 } M _ { \odot } ) for models with with \sigma _ { 8 } = 0.9 and V _ { c } > 35 km/s ( M _ { vir } = ( 6 - 8 ) \times 10 ^ { 9 } M _ { \odot } ) for \sigma _ { 8 } = 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 being too small as compared with the observations .
The problem is that many of the observed dwarf galaxies 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 .
We also estimate the rms deviations from the Hubble flow \sigma _ { H } for galaxies at different distances from the Local Group and find that in most of our model LV-candidates the rms peculiar velocities are consistent with observational values : \sigma _ { H } = 50 km/s for distances less than 3 Mpc and \sigma _ { 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 ) – significantly larger than typically assumed .