Knowledge of the interior structure of Mars is of fundamental importance to the understanding of its past and present state as well as its future evolution. The most prominent interior structure properties are the state of the core, solid or liquid, its radius, and its composition in terms of light elements, the thickness of the mantle, its composition, the presence of a lower mantle, and the density of the crust. In the absence of seismic sounding only geodesy data allow reliably constraining the deep interior of Mars. Those data are the mass, moment of inertia, and tides. They are related to Mars' composition, to its internal mass distribution, and to its deformational response to principally the tidal forcing of the Sun. Here we use the most recent estimates of the moment of inertia and tidal Love number k2 in order to infer knowledge about the interior structure of the Mars. We have built precise models of the interior structure of Mars that are parameterized by the crust density and thickness, the volume fractions of upper mantle mineral phases, the bulk mantle iron concentration, and the size and the sulfur concentration of the core. From the bulk mantle iron concentration and from the volume fractions of the upper mantle mineral phases, the depth dependent mineralogy is deduced by using experimentally determined phase diagrams. The thermoelastic properties at each depth inside the mantle are calculated by using equations of state. Since it is difficult to determine the temperature inside the mantle of Mars we here use two end-member temperature profiles that have been deduced from studies dedicated to the thermal evolution of Mars. We calculate the pressure and temperature dependent thermoelastic properties of the core constituents by using equations state and recent data about reference thermoelastic properties of liquid iron, liquid iron-sulfur, and solid iron. To determine the size of a possible inner core we use recent data on the melting temperature of iron-sulfur. Within our model assumptions the geodesy data imply that Mars has no solid inner core and that the liquid core contains a large fraction of sulfur. The absence of a solid inner is in agreement with the absence of a global magnetic field. We estimate the radius of the core to be 1794 ± 65 km and its core sulfur concentration to be 16 ± 2 wt%. We also show that it is possible for Mars to have a thin layer of perovskite at the bottom of the mantle if it has a hot mantle temperature. Moreover a chondritic Fe/Si ratio is shown to be consistent with the geodesy data, although significantly different value are also possible. Our results demonstrate that geodesy data alone, even if a mantle temperature is assumed, can almost not constrain the mineralogy of the mantle and the crust. In order to obtain stronger constraints on the mantle mineralogy bulk properties, like a fixed Fe/Si ratio, have to be assumed.