  Model reduction of the compressible Euler equations based on proper
orthogonal decomposition (POD) and Galerkin orthogonality or least-squares
residual minimization requires the selection of inner product spaces in which
to perform projections and measure norms. The most popular choice is the
vector-valued L2({\Omega}) inner product space. This choice, however, yields
dimensionally-inconsistent reduced-order model (ROM) formulations which often
lack robustness. In this work, we try to address this weakness by studying a
set of dimensionally-consistent inner products with application to the
compressible Euler equations. First, we demonstrate that non-dimensional inner
products have a positive impact on both POD and Galerkin/least-squares ROMs.
Second, we further demonstrate that physics-based inner products based on
entropy principles result in drastically more accurate and robust ROM
formulations than those based on non-dimensional L2({\Omega}) inner products.As
test cases, we consider the following well-known problems: the one-dimensional
Sod shock tube, the two-dimensional Kelvin-Helmholtz instability and
two-dimensional homogeneous isotropic turbulence.
