  The angular diameter distances toward galaxy clusters can be determined with
measurements of the Sunyaev-Zel'dovich effect and X-ray surface brightness
combined with the validity of the distance-duality relation, $D_L(z) (1 +
z)^{2}/D_{A}(z) = 1$, where $D_L(z)$ and $D_A(z)$ are, respectively, the
luminosity and angular diameter distances. This combination enables us to probe
galaxy cluster physics or even to test the validity of the distance-duality
relation itself. We explore these possibilities based on two different, but
complementary approaches. Firstly, in order to constrain the possible galaxy
cluster morphologies, the validity of the distance-duality relation (DD
relation) is assumed in the $\Lambda$CDM framework (WMAP7). Secondly, by
adopting a cosmological-model-independent test, we directly confront the
angular diameters from galaxy clusters with two supernovae Ia (SNe Ia)
subsamples (carefully chosen to coincide with the cluster positions). The
influence of the different SNe Ia light-curve fitters in the previous analysis
are also discussed. We assumed that $\eta$ is a function of the redshift
parametrized by two different relations: $\eta(z) = 1 + \eta_{0}z$, and
$\eta(z)=1 + \eta_{0}z/(1+z)$, where $\eta_0$ is a constant parameter
quantifying the possible departure from the strict validity of the DD relation.
The statistical analysis presented here provides new evidence that the true
geometry of clusters is elliptical. We find that the two-light curve fitters
(SALT2 and MLCS2K2) present a statistically significant conflict, and a joint
analysis involving the different approaches suggests that clusters are endowed
with an elliptical geometry as previously assumed.
