Large Magellanic Cloud Near-Infrared Synoptic Survey. II. The Wesenheit
  relations and their application to the Distance scale
  We present new near-infrared Cepheid Period-Wesenheit relations in the LMC
using time-series observations from the Large Magellanic Cloud Near-Infrared
Synoptic Survey. We also derive optical$+$near-infrared P-W relations using $V$
and $I$~magnitudes from OGLE-III. We employ our new $JHK_s$ data to determine
an independent distance to the LMC of $\mu_{\rm LMC} = 18.47\pm0.07
{\textit{(statistical)}}$~mag, using an absolute calibration of the Galactic
relations based on several distance determination methods and accounting for
the intrinsic scatter of each technique. We also derive new near-infrared
Period-Luminosity and Wesenheit relations for Cepheids in M31 using
observations from the PHAT survey. We use the absolute calibrations of the
Galactic and LMC $W_{J,H}$ relations to determine the distance modulus of M31,
$\mu_{\rm M31} = 24.46\pm0.20$~mag. We apply a simultaneous fit to Cepheids in
several Local Group galaxies covering a range of metallicities
($7.7<12+\log[O/H]<8.6$~dex) to determine a global slope of
-$3.244\pm0.016$~mag/dex for the $W_{J,K_s}$ relation and obtain robust
distance estimates. Our distances are in good agreement with recent TRGB based
distance estimates and we do not find any evidence for a metallicity dependence
in the near-infrared P-W relations.
