Convex rank 1 subsets of Euclidean buildings (of type A 2)

For a Euclidean building X of type A ₂, we classify the 0-dimensional subbuildings A of ∂ _T X that occur as the asymptotic boundary of closed convex subsets. In particular, we show that triviality of the holonomy of a triple (of points of A) is (essentially) sufficient. To prove this, we construct new convex subsets as the union of convex sets.