The notion of convexity in tropical geometry is closely related to notions of
convexity in the theory of affine buildings. We explore this relationship from
a combinatorial and computational perspective. Our results include a convex
hull algorithm for the Bruhat--Tits building of SLd​(K) and techniques for
computing with apartments and membranes. While the original inspiration was the
work of Dress and Terhalle in phylogenetics, and of Faltings, Kapranov, Keel
and Tevelev in algebraic geometry, our tropical algorithms will also be
applicable to problems in other fields of mathematics.Comment: 22 pages, 4 figure