  We construct entropy conservative and entropy stable high order accurate
discontinuous Galerkin (DG) discretizations for time-dependent nonlinear
hyperbolic conservation laws on curvilinear meshes. The resulting schemes
preserve a semi-discrete quadrature approximation of a continuous global
entropy inequality. The proof requires the satisfaction of a discrete geometric
conservation law, which we enforce through an appropriate polynomial
approximation. We extend the construction of entropy conservative and entropy
stable DG schemes to the case when high order accurate curvilinear mass
matrices are approximated using low-storage weight-adjusted approximations, and
describe how to retain global conservation properties under such an
approximation. The theoretical results are verified through numerical
experiments for the compressible Euler equations on triangular and tetrahedral
meshes.
