  (abridged) The aim of this paper is to investigate the metallicity dependence
of the PL relation in V and K, based on a sample of 128 Galactic, 36 LMC, and 6
SMC Cepheids with individual Baade-Wesselink (BW) distances and individually
determined metallicities from high-resolution spectroscopy. The p-relation
finally adopted is 1.50 -0.24log P. The slope of this relation is based on the
condition that the distance to the LMC does not depend on period or (V-K)
colour and that the slope of the PL relation based on the BW distances agrees
with that based on apparent magnitude. The zero point of the relation is tight
to the Cepheids with HST and revised Hipparcos parallaxes as well as to
Cepheids in clusters. The slope of the Galactic and LMC K-band relation
formally agrees within the errors, and combining all Cepheids (including the
SMC) results in a negligible metallicity dependence. A similar conclusion is
found for the reddening-free Wesenheit relation. In the V-band the situation is
more complex. The slope of the LMC and the Galactic PL relation differ at the
3sigma level. Combining the sample nevertheless results in a metallicity term
significant at the 2sigma level. The details of the comparison of BW-based
distances and Cepheids with HST and revised Hipparcos parallaxes also play a
role. The method used by Storm et al. would lead to larger DM of 18.37 and
18.81 for the LMC and SMC, respectively. The LMC DM is shorter than the
currently accepted value, which is in the range 18.42 to 18.55 (Walker 2012),
and it is speculated that the p-factor may depend on metallicity.
