Local convexity in CAT(\kappa)-spaces
  Heinrich Tietze has shown that for a closed connected subset of euclidean space being convex is a local property. We generalize this to CAT(0)-spaces and locally compact CAT(\kappa) spaces. As an application we give a construction of certain convex sets in euclidean buildings. 
Kai-Uwe Bux, Stefan Witzel
math.MG