Context : The metallicity dependence of the Cepheid period-luminosity ( PL ) relation is of importance in establishing the extra-galactic distance scale .

Aims : 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 Large Magellanic Cloud ( LMC ) , and 6 Small Magellanic Cloud ( SMC ) Cepheids with individual Baade-Wesselink ( BW ) distances ( some of the stars also have an Hubble Space Telescope ( HST ) based and Hipparcos parallax or are in clusters ) and individually determined metallicities from high-resolution spectroscopy .

Methods : Literature values of the V -band , K -band , and radial velocity data were collected for the sample of Cepheids .
Based on a ( V - K ) surface-brightness relation and a projection factor , distances were derived from a BW analysis .

Results : The p -relation finally adopted is 1.50 - 0.24 \log 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 and a relation of M _ { K } = ( -2.50 \pm 0.08 ) + ( -3.06 \pm 0.06 ) \log P .
A similar conclusion is found for the reddening-free Wesenheit relation ( W ( VK ) = K - 0.13 ( V - K ) ) , with M _ { WVK } = ( -2.68 \pm 0.08 ) + ( -3.12 \pm 0.06 ) \log P .
In the V -band the situation is more complex .
The slope of the LMC and the Galactic PL relation differ at the 3 \sigma level .
Combining the sample nevertheless results in a metallicity term significant at the 2 \sigma level : M _ { V } = ( -1.55 \pm 0.09 ) + ( -2.33 \pm 0.07 ) \log P + ( +0.23 \pm 0.11 ) [ Fe/H ] .
Taking only the Galactic Cepheids , the metallicity term is no longer significant , namely ( +0.17 \pm 0.25 ) .
Compared to the recent works by Storm et al .
( 2011a , b ) , there is both agreement and disagreement .
A similar dependence of the p -factor on period is found , but the zero point found here implies a shorter distance scale .
The distance modulus ( DM ) to the LMC and SMC found here are 18.29 \pm 0.02 and 18.73 \pm 0.06 ( statistical error on the mean ) , respectively .
Systematic differences in reddening could have an effect of order +0.05 in DM .
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 .
This is not predicted by theoretical investigations , but these same investigations do not predict a steep dependence on period either , indicating that additional theoretical work is warranted .

Conclusions :