MV-polytopes via affine buildings

For an algebraic group G, Anderson introduced the notion of Mirković-Vilonen (MV) polytopes as images of MV-cycles under the moment map of the affine Grassmannian. It was shown by Kamnitzer that MV-polytopes and their corresponding cycles can be described as solutions of the tropical Plücker relations. Another construction of MV-cycles, by Gaussent and Littelmann, can be given by using LS-galleries, a more discrete version of Littelmann's path model. This article gives a direct combinatorial construction of the MV-polytopes using LS-galleries. This construction is linked to the retractions of the affine building and the Bott-Samelson variety corresponding to G, leading to a type-independent definition of MV-polytopes not involving the tropical Plücker relations.