  We carry out a study of the global three-dimensional (3D) structure of the
electron density and temperature of the quiescent inner solar corona ($r<1.25
R_\odot$) by means of tomographic reconstructions and magnetohydrodynamic
simulations. We use differential emission measure tomography (DEMT) and the
Alfv\'en Wave Solar Model (AWSoM), in their latest versions. Two target
rotations were selected from the solar minimum between solar cycles (SCs) 23
and 24 and the declining phase of SC 24. We report in quantitative detail on
the 3D thermodynamic structure of the core and outer layers of the streamer
belt, and of the high latitude coronal holes (CH), as revealed by the DEMT
analysis. We report on the presence of two types of structures within the
streamer belt, loops with temperature decreasing/increasing with height (dubbed
down/up loops), as reported first in previous DEMT studies. We also estimate
the heating energy flux required at the coronal base to keep these structures
stable, found to be or order $10^5 erg\, cm^{-2} s^{-1}$, consistently with
previous DEMT and spectroscopic studies. We discuss how these findings are
consistent with coronal dissipation of Alfv\'en waves. We compare the 3D
results of DEMT and AWSoM in distinct magnetic structures. We show that the
agreement between the products of both techniques is the best so far, with an
overall agreement $\lesssim 20\%$, depending on the target rotation and the
specific coronal region. In its current implementation the ASWsoM model can not
reproduce down loops though. Also, in the source region of the fast and slow
components of the solar wind, the electron density of the AWSoM model increases
with latitude, opposite to the trend observed in DEMT reconstructions.
