2008/11/07 by Jack Girolo, Jack E. Girolo, Girolo, Jack E.
Mathematics · #Advanced Topology and Set Theory #Functional Equations Stability Results #Mathematics and Applications #math.GN #math.MG #msc:53C23 #msc:54C55 #msc:54E50
paper · pdf · doi:10.48550/arxiv.0811.1227
22 pages, 2 figures
arxiv created 2008/11/07 · arxiv updated 2009/12/01
Let (X, d) be a Cat(k) space and P a bounded subset of X . If k > 0 then it is required that the diameter of P be less than Pi/(4 sqrt(k)) . Let u: P to R be a bounded non-negative function from P to R. The existence of a unique point in X called the barycenter of P relative to u is established. When u=1, the barycenter is simply the circumcenter of P. The barycenter has a number of properties including a scaling, continuity and limit property. Under suitable conditions, the barycenter is a fixed point of an isometry or group of isometries. Barycenters are used to show that a complete Cat(k) space X is an absolute retract if k is less than or equal to 0, and an absolute neighborhood retract if X is complete and of curvature less than or equal to k.