A subset \(C\) of a metric space \((X,d)\) is a Chebyshev set if each point in \(X\) has a unique closest point in \(C\). With the added structure for \((X,d)\) being a normed linear space questions involving the convexity of \(C\) become central (as well as questions about the smoothness of the unit ball of \(X\)). It is an open question if every Chebyshev set in a Hilbert space (even in a separable Hilbert space) is convex. It is known that under a variety of conditions, many related to compactness (e.g., \(X\) is finite dimensional) that \(C\) is convex. There also exists an example of a non-convex Chebyshev set in an incomplete, inner-product space.
This paper proves and catalogs analogous conditions related to the convexity of Chebyshev sets when \((X,d)\) is generalized from a linear space to a geodesic metric spaces with bounded curvature.
\((X,d)\) is a geodesic space if, for every \(x\) and \(y\) in \(X\), there is a unique continuous mapping (called the geodesic connecting \(x\) and \(y\)), \(c(t)\), of \([0,1]\) into \(X\) such that \(c(0) = x\), \(c(1) = y\) and \(d(c(s), c(t)) = |t-s|\) for \(0 \leq s \leq t \leq 1\).
A set, \(C\), is called convex if for every \(x\) and \(y\) in \(C\), \(C\) contains the geodesic connecting \(x\) and \(y\). A condition on triangles composed of geodesics defines bounded curvature of \((X,d)\).