  This paper studies best finitely supported approximations of one-dimensional
probability measures with respect to the $L^r$-Kantorovich (or transport)
distance, where either the locations or the weights of the approximations'
atoms are prescribed. Necessary and sufficient optimality conditions are
established, and the rate of convergence (as the number of atoms goes to
infinity) is discussed. In view of emerging mathematical and statistical
applications, special attention is given to the case of best uniform
approximations (i.e., all atoms having equal weight). The approach developed in
this paper is elementary; it is based on best approximations of (monotone)
$L^r$-functions by step functions, and thus different from, yet naturally
complementary to, the classical Voronoi partition approach.
