Galaxy clusters are the most massive gravitationally bound systems
known. Their population provides most valuable cosmological
information. To collect this information, clusters need to be
physically characterized based on an understanding of their
complicated structures. Galaxy clusters are predominantly composed by
galaxies, by the intercluster medium (ICM) gas which fills the space
between the galaxy members, and by dark matter.
In virialized clusters, the ICM gas is heated to about 10^7 Kelvin and
can be studied via its diffuse X-ray emission and via the thermal
Sunyaev-Zel'dovich (tSZ) effect. Assuming hydrostatic equilibrium, the
study of this ICM gas can reveal the depth and structure of the
cluster potential that is shaped by the presence of dark matter.
Alternatively, cluster gravitational potentials can be inferred from
gravitational lensing, which is insensitive of equilibrium
assumptions. Consequently, the joint analysis of different cluster
observables provides complementary information on the physical state
of the cluster gas and on the cluster geometry.

I will present a method to reconstruct the projected gravitational
potential of galaxy clusters from observations of the ICM gas
(hereafter "HE potential"). This method adopts a polytropic relation
for the stratification of the ICM, assumes hydrostatic equilibrium and
a simple (but not necessarily spherical) cluster geometry. I will
first discuss these assumptions. Then, given that gravitational
potentials are local quantities, I will illustrate how the comparison
of the HE potentials with the lensing potential may help us to probe
locally the physical state of the cluster. Finally, I will show a
method to reconstruct the projected potentials jointly constrained by
X-ray and weak lensing observations.