  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.
