  We motivate the study of metric spaces with a unique convex geodesic
bicombing, which we call CUB spaces. These encompass many classical notions of
nonpositive curvature, such as CAT(0) spaces and Busemann-convex spaces. Groups
having a geometric action on a CUB space enjoy numerous properties.
  We want to know when a simplicial complex, endowed with a natural polyhedral
metric, is CUB. We establish a link condition, stating essentially that the
complex is locally a lattice. This generalizes Gromov's link condition for cube
complexes, for the $\ell^\infty$ metric.
  The link condition applies to numerous examples, including Euclidean
buildings, simplices of groups, Artin complexes of Euclidean Artin groups,
(weak) Garside groups, some arcs and curve complexes, and minimal spanning
surfaces of knots.
