The relative Chebyshev centers of finite sets in geodesic spaces (Q1006959)

A metric space \(X\) is a \textit{geodesic~space} if any two points \(x,y\) of \(X\) may be joined by a continuous curve whose length equals the distance \(d(x,y)\). If \(M\) is a bounded subset of \(X\), then its \textit{Chebychev~radius} \(R(M)\) is the infimum of the radii \(R\) of all metric balls \(B(x,R)\) that contain \(M\), where \(x\) ranges over all points of \(X\). A point \(x\) of \(X\) is a \textit{Chebychev~center} of \(M\) if \(M\) is contained in \(B(x,R(M))\). Let \(Z(M)\) denote the set of all Chebychev centers of \(M\). One may also define relative versions \(R_{W}(M)\) and \(Z_{W}(M)\), where \(W\) is a bounded subset of \(M\) and the point \(x\) in the definitions above ranges over \(W\) instead of \(X\). The collection of bounded subsets of \(X\) is a metric space with respect to the Hausdorff metric. In this paper the author considers finite or compact subsets \(M\) of \(X\) and describes how \(R(M)\) and \(Z(M)\) vary as \(M\) varies. There are three main results, two of which assume that the metric space \(X\) has convex metric balls and unique shortest geodesic segments between any two points. These extra properties are satisfied by complete, simply connected, Riemannian manifolds of nonpositive sectional curvature. The main results are somewhat complicated to state, and we omit a precise formulation.