  A Newton-Okounkov convex body is a convex body constructed from a projective
variety with a valuation on its homogeneous coordinate ring; this is deeply
connected with representation theory. For instance, the Littelmann string
polytopes and the Feigin-Fourier-Littelmann-Vinberg polytopes are examples of
Newton-Okounkov convex bodies. In this paper, we prove that the Newton-Okounkov
convex body of a Schubert variety with respect to a specific valuation is
identical to the Nakashima-Zelevinsky polyhedral realization of a Demazure
crystal. As an application of this result, we show that Kashiwara's involution
(*-operation) corresponds to a change of valuations on the rational function
field.
