On a new method to estimate distance, reddening and metallicity of RR
  Lyrae stars using optical/near-infrared ($B$,$V$,$I$,$J$,$H$,$K$) mean
  magnitudes: $\omega$ Centauri as a first test case
  We developed a new approach to provide accurate estimates of metal content,
reddening and true distance modulus of RR Lyrae stars (RRLs). The method is
based on homogeneous optical ($BVI$) and near-infrared ($JHK$) mean magnitudes
and on predicted period--luminosity--metallicity relations ($IJHK$) and
absolute mean magnitude--metallicity relations ($BV$). We obtained solutions
for three different RRL samples in $\omega$ Cen: first overtone (RRc,~90),
fundamental (RRab,~80) and global (RRc+RRab) in which the period of first
overtones were fundamentalized. The metallicity distribution shows a well
defined peak at [Fe/H]$\sim$--1.98 and a standard deviation of $\sigma$=0.54
dex. The spread is, as expected, metal-poor ([Fe/H]$\le$--2.3) objects. The
current metallicity distribution is $\sim$0.3 dex more metal-poor than similar
estimates for RRLs available in the literature. The difference vanishes if the
true distance modulus we estimated is offset by --0.06/--0.07~mag in true
distance modulus. We also found a cluster true distance modulus of
$\mu$=13.720$\pm$0.002$\pm$0.030~mag, where the former error is the error on
the mean and the latter is the standard deviation. Moreover, we found a cluster
reddening of E($B-V$)=0.132$\pm$0.002$\pm$0.028~mag and spatial variations of
the order of a few arcmin across the body of the cluster. Both the true
distance modulus and the reddening are slightly larger than similar estimates
available in the literature, but the difference is within 1$\sigma$. The
metallicity dependence of distance diagnostics agree with theory and
observations, but firm constraints require accurate and homogeneous
spectroscopic measurements.
