Polygons with prescribed Gauss map in Hadamard spaces and Euclidean buildings

We show that given a stable weighted configuration on the asymptotic boundary of a locally compact Hadamard space, there is a polygon with Gauss map prescribed by the given weighted configuration if the configuration is stable. Moreover, the same result holds for semistable configurations on arbitrary Euclidean buildings. We also examine how rays project to closed convex subsets of Hadamard spaces.