Physical properties of the ambient medium and of dense cores in the Perseus star-forming region derived from Herschel Gould Belt Survey observations.

(Abridged) In this paper, we present analyses of images taken with the Herschel ESA satellite from 70mu to 500mu. We first constructed column density and dust temperature maps. Next, we identified compact cores in the maps, and characterize the cores using modified blackbody fits to their SEDs: we identified 684 starless cores, of which 199 are bound and potential prestellar cores, and 132 protostars. We also matched the Herschel-identified young stars with GAIA sources to model distance variations across the Perseus cloud. We measure a linear gradient function with right ascension and declination for the entire cloud. From the SED fits, mass and temperature of cores were derived. The core mass function can be modelled with a log-normal distribution that peaks at 0.82~$M_\sun$ suggesting a star formation efficiency of 0.30. The high-mass tail can be modelled with a power law of slope $\sim-2.32$, close to the Salpeter's value. We also identify the filamentary structure of Perseus, confirming that stars form preferentially in filaments. We find that the majority of filaments where star formation is ongoing are transcritical against their own internal gravity because their linear masses are below the critical limit of 16~$M_\sun$pc$^{-1}$ above which we expect filaments to collapse. We find a possible explanation for this result, showing that a filament with a linear mass as low as 8~$M_\sun$pc$^{-1}$ can be already unstable. We confirm a linear relation between star formation efficiency and slope of dust probability density function and a similar relation is also seen with the core formation efficiency. We derive a lifetime for the prestellar core phase of $1.69\pm0.52$~Myr for Perseus but different regions have a wide range in prestellar core fractions, hint that star-formation has started only recently in some clumps. We also derive a free-fall time for prestellar cores of 0.16~Myr.