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We show, using a direct variational approach, that the second boundary valueproblem for the Monge-Amp\`ere equation in R^n with exponential non-linearityand target a convex body P is solvable iff 0 is the barycenter of P. Combinedwith some toric geometry this confirms, in particular, the (generalized)Yau-Tian-Donaldson conjecture for toric log Fano varieties (X,D), saying that(X,D) admits a (singular) K\"ahler-Einstein metric iff it is K-stable in thealgebro-geometric sense. We thus obtain a new proof and extend to the log Fanosetting the seminal result of Zhou-Wang concerning the case when X is smoothand D is trivial. Li's toric formula for the greatest lower bound on the Riccicurvature is also generalized. More generally, we obtain K\"ahler-Riccisolitons on any log Fano variety and show that they appear as the large timelimit of the K\"ahler-Ricci flow. Furthermore, using duality, we also confirm aconjecture of Donaldson concerning solutions to Abreu's boundary value problemon the convex body P. in the case of a given canonical measure on the boundaryof P.

Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties