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Proximinality in geodesic spaces. (English) Zbl 1141.51011
A geodesic metric space is a CAT(0) space if it is geodesically connected, and all distances between points on the sides of a geodesic triangle are no larger than the distances between the corresponding points on any comparison triangle in the Euclidean plane. It is previously known that for any compact set C in a complete length space X without bifurcating geodesics, the nearest point projection of X onto C is properly single valued at most points of X. The authors show that the same is true for a closed subset of a complete CAT(0) space X, where X has the geodesic extension property and has Alexandrov curvature bounded below. The authors also show that if C is bounded and closed, then the set of points of X which have a unique farthest point in C is dense in X.

MSC:
51F99 Metric geometry
51K05 General theory of distance geometry
53C22 Geodesics in global differential geometry
47H10 Fixed-point theorems
46B20 Geometry and structure of normed linear spaces
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