Reflexivity of Newton--Okounkov bodies of partial flag varieties

Assume that the valuation semigroup \Gamma (\lambda ) of an arbitrary partial flag variety corresponding to the line bundle \mathcal {L_\lambda } constructed via a full-rank valuation is finitely generated and saturated. We use Ehrhart theory to prove that the associated Newton–Okounkov body — which happens to be a rational, convex polytope — contains exactly one lattice point in its interior if and only if \mathcal {L}_\lambda is the anticanonical line bundle. Furthermore, we use this unique lattice point to construct the dual polytope of the Newton–Okounkov body and prove that this dual is a lattice polytope using a result by Hibi. This leads to an unexpected, necessary and sufficient condition for the Newton–Okounkov body to be reflexive.