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The Calabi theorem states that for every regular convex cone K in R^n, the Monge-Ampère equation log det F” = 2F/n has a unique convex solution on the interior of K which tends to +infty on the boundary of K. It turns out that this solution is self-concordant and logarithmically homogeneous, and thus is a barrier which can be used for conic optimization. We consider different aspects of this barrier: