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This paper designs and analyzes positivity-preserving well-balanced (WB) central discontinuous Galerkin (CDG) schemes for the Euler equations with gravity. A distinctive feature of these schemes is that they not only are WB for a general known stationary hydrostatic solution, but also can preserve the positivity of the fluid density and pressure. The standard CDG method does not possess this feature, while directly applying some existing WB techniques to the CDG framework may not accommodate the positivity and keep other important properties at the same time. In order to obtain the WB and positivity-preserving properties simultaneously while also maintaining the conservativeness and stability of the schemes, a novel spatial discretization is devised in the CDG framework based on suitable modifications to the numerical dissipation term and the source term approximation. The modifications are based on a crucial projection operator for the stationary hydrostatic solution, which is proposed for the first time in this work. This novel projection has the same order of accuracy as the standard $L^2$-projection, can be explicitly calculated, and is easy to implement without solving any optimization problems. More importantly, it ensures that the projected stationary solution has the same cell averages on both the primal and dual meshes, which is a key to achieve the desired properties of our schemes. Based on some convex decomposition techniques, rigorous positivity-preserving analyses for the resulting WB CDG schemes are carried out. Several one- and two-dimensional numerical examples are performed to illustrate the desired properties of these schemes, including the high-order accuracy, the WB property, the robustness for simulations involving the low pressure or density, high resolution for the discontinuous solutions and the small perturbations around the equilibrium state.