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We show that any compact convex simple lattice polytope is the momentpolytope of a K\"ahler-Einstein orbifold, unique up to orbifold covering andhomothety. We extend the Wang-Zhu Theorem \cite{WZ} giving the existence of aK\"ahler-Ricci soliton on any toric monotone manifold on any compact convexsimple labelled polytope satisfying the combinatoric condition corresponding tomonotonicity. We obtain that any compact convex simple polytope $P\subset\bR^n$ admits a set of inward normals, unique up to dilatation, such that thereexists a symplectic potential satisfying the Guillemin boundary condition (withrespect to these normals) and the K\"ahler-Einstein equation on $P\times\bR^n$. We interpret our result in terms of existence of singularK\"ahler-Einstein metrics on toric manifolds.

Toric K\"ahler-Einstein metrics and convex compact polytopes