Monge–Ampère Equation With Guillemin Boundary Condition in High Dimension

ABSTRACT The Guillemin boundary condition naturally appears in the study of Kähler geometry of toric manifolds. In the present paper, the following Guillemin boundary value problem is investigated: , where and is a simple convex polytope in . The solvability of this problem is given under the necessary and sufficient condition. The key issue in the proof is to obtain the boundary regularity of . Due to the difficulty caused by the structure of the equation itself and the singularity of , special attention is required to understand the influence of different singularity types at various positions on and how these impact the behavior of in its vicinity.