A Generalization of the Circumcenter of a Set

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.

💡 Research Summary

The paper investigates a natural generalisation of the circumcenter in the setting of metric spaces of bounded curvature, i.e. CAT(k) spaces. Let (X,d) be a CAT(k) space and let P⊂X be a bounded subset. When k>0 the authors impose the standard diameter restriction diam P < π/(4√k) to guarantee the necessary convexity properties. A bounded non‑negative function u:P→ℝ is introduced; it can be thought of as assigning a weight or “penalty” to each point of P. The central object of the study is the barycenter of P relative to u, defined as the unique point x∈X that minimises the supremum‑type functional

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