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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 is deeplyconnected with representation theory. For instance, the Littelmann stringpolytopes and the Feigin-Fourier-Littelmann-Vinberg polytopes are examples ofNewton-Okounkov convex bodies. In this paper, we prove that the Newton-Okounkovconvex body of a Schubert variety with respect to a specific valuation isidentical to the Nakashima-Zelevinsky polyhedral realization of a Demazurecrystal. As an application of this result, we show that Kashiwara's involution(*-operation) corresponds to a change of valuations on the rational functionfield.

Newton-Okounkov convex bodies of Schubert varieties and polyhedral realizations of crystal bases