  Geodesic metric spaces support a variety of averaging constructions for given
finite sets. Computing such averages has generated extensive interest in
diverse disciplines. Here we consider the inverse problem of recognizing
computationally whether or not a given point is such an average, exactly or
approximately. In nonpositively curved spaces, several averaging notions,
including the usual weighted barycenter, produce the same "mean set". In such
spaces, at points where the tangent cone is a Euclidean space, the recognition
problem reduces to Euclidean projection onto a polytope. Hadamard manifolds
comprise one example. Another consists of CAT(0) cubical complexes, at
relative-interior points: the recognition problem is harder for general points,
but we present an efficient semidefinite-programming-based algorithm.
