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We show that lattice polytopes cut out by root systems of classical type arenormal and Koszul, generalizing a well-known result of Bruns, Gubeladze, andTrung in type A. We prove similar results for Cayley sums of collections ofpolytopes whose Minkowski sums are cut out by root systems. The proofs arebased on a combinatorial characterization of diagonally split toric varieties.

Lattice polytopes cut out by root systems and the Koszul property