  The number density of field galaxies per rotation velocity, referred to as
the velocity function, is an intriguing statistical measure probing the
smallest scales of structure formation. In this paper we point out that the
velocity function is sensitive to small shifts in key cosmological parameters
such as the amplitude of primordial perturbations ($\sigma_8$) or the total
matter density ($\Omega_{\rm m}$). Using current data and applying conservative
assumptions about baryonic effects, we show that the observed velocity function
of the Local Volume favours cosmologies in tension with the measurements from
Planck but in agreement with the latest findings from weak lensing surveys.
While the current systematics regarding the relation between observed and true
rotation velocities are potentially important, upcoming data from HI surveys as
well as new insights from hydrodynamical simulations will dramatically improve
the situation in the near future.
