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

Newton-Okounkov bodies for Bott-Samelson varieties and string polytopes for generalized Demazure modules