  A Newton-Okounkov convex body is a convex body constructed from a projective
variety with a valuation on its homogeneous coordinate ring; this generalizes a
Newton polytope for a toric variety. This convex body has various kinds of
information about the original projective variety; for instance, Kaveh showed
that the string polytopes from representation theory are examples of
Newton-Okounkov bodies for Schubert varieties. In this paper, we extend the
notion of string polytopes for Demazure modules to generalized Demazure
modules, and prove that the resulting generalized string polytopes are
identical to the Newton-Okounkov bodies for Bott-Samelson varieties with
respect to a specific valuation. As an application of this result, we show that
these are indeed polytopes.
