We prove an analogue of Kostants convexity theorem for thick affine buildings and give an application for groups with affine BN -pair. Recall that there are two natural retractions of the affine building onto a fixed apartment A: The retraction r centered at an alcove in A and the retraction ρ centered at a chamber in the spherical building at infinity. We prove that for each special vertex X ∈ A the set ρ(r −1 (W.x)) is a certain convex hull of W.x. The proof can be reduced to a problem in

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... ter complexes and heavily relies on a character formula for highest weight representations of algebraic groups.