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The affine hyperspheres can be described as the level sets of solutions to the \"hyperbolic\" toric K\\\"ahler-Einstein equation $e^{\\Phi} = \\det D^2 \\Phi$ on proper convex cones. We prove a generalization of this theorem by showing that for every\n  $\\Phi$ solving this equation on a proper convex domain $\\Omega$ the corresponding metric measure space $(D^2 \\Phi, e^{\\Phi}dx)$ has a non-positive Bakry-{\\'E}mery tensor. 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