Okounkov bodies and toric degenerations
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Let \(\varDelta \) be the Okounkov body of a divisor \(D\) on a projective variety \(X\). We describe a geometric criterion for \(\varDelta \) to be a lattice polytope, and show that in this situation \(X\) admits a flat degeneration to the corresponding toric variety. This degeneration is functorial in an appropriate sense.