  We give a metric characterization of the scalar curvature of a smooth
Riemannian manifold, analyzing the maximal distance between $(n+1)$ points in
infinitesimally small neighborhoods of a point. Since this characterization is
purely in terms of the distance function, it could be used to approach the
problem of defining the scalar curvature on a non-smooth metric space. In the
second part we will discuss this issue, focusing in particular on Alexandrov
spaces and surfaces with bounded integral curvature.
