Concave transforms of filtrations and rationality of Seshadri constants

We show that the subgraph of the concave transform of a multiplicative filtration on a section ring is the Newton--Okounkov body of a certain semigroup, and if the filtration is induced by a divisorial valuation, then the associated graded algebra is the algebra of sections of a concrete line bundle in higher dimension. We use this description to give a rationality criterion for certain Seshadri constants. Along the way we introduce Newton--Okounkov bodies of abstract graded semigroups and determine conditions for their slices to be Newton--Okounkov bodies of subsemigroups.