Not yet reviewed by Pith; the record is open.

Volume density is a key physical quantity controlling the evolution of the interstellar medium (ISM) and star formation, but it cannot be accessed directly by observations of molecular clouds. We aim at estimating the volume density distribution in nearby molecular clouds, to measure the relation between column and volume densities and to determine their roles as predictors of star formation. We develop an inverse modelling method to estimate the volume density distributions of molecular clouds. We apply this method to 24 nearby molecular clouds for which column densities have been derived using Herschel observations and for which star formation efficiencies (SFE) have been derived using observations with the Spitzer space telescope. We then compare the relationships of several column- and volume-density based descriptors of dense gas with the SFE of the clouds. We derive volume density distributions for 24 nearby molecular clouds, which represents the most complete sample of such distributions to date. The relationship between column densities and peak volume densities in these clouds is a piece-wise power-law relation that changes its slope at a column density of $5-10\times 10^{22}$ H$_2$cm$^{-2}$. We interpret this as a signature of hierarchical fragmentation in the dense ISM. We find that the volume-density based dense gas fraction is the best predictor of star formation in the clouds, and in particular, it is as anticipated a better predictor than the column-density based dense gas fraction. We also derive a volume density threshold density for star formation of $2\times 10^4$ H$_2$cm$^{-3}$.