Connecting toric manifolds by conical Kähler-Einstein metrics

We give criterions for the existence of toric conical Kähler-Einstein and Kähler-Ricci soliton metrics on any toric manifold in relation to the greatest Ricci and Bakry-Emery-Ricci lower bound. We also show that any two toric manifolds with the same dimension can be joined by a continuous path of toric manifolds with conical Kähler-Einstein metrics in the Gromov-Hausdorff topology.