A Generalization of the Circumcenter of a Set

TL;DR

This paper generalizes the concept of the circumcenter in Cat(k) spaces by introducing barycenters relative to a function, establishing their properties and applications in geometric topology.

Contribution

It introduces a new notion of barycenter relative to a function in Cat(k) spaces, extending the classical circumcenter concept and exploring its properties and topological implications.

Findings

Existence and uniqueness of barycenters in Cat(k) spaces.

Barycenters have scaling, continuity, and limit properties.

Applications to fixed points and topological retracts in geometric spaces.

Abstract

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.