Communicated by R. R. Goldberg, February 25, 1977 1. Introduction and results. Let S N C C[0, 1] denote the class of all piecewise linear functions with at most TV + 1 linear segments. In this article we announce some interesting and somewhat surprising approximation properties of S N in the space L 2 [0, 1]. Three main theorems will be stated in this section and the main idea of our proof of the first two theorems will be sketched in Β§2.Theorem 1 describes a fairly large class of strictly convex functions which have, for each positive integer N, unique best L 2 [0, 1] approximants from the nonlinear (spline) manifold S N . Theorem 2 states that any sufficiently smooth strictly convex function eventually, i.e. for all large N, has a unique best L 2 [0,1] approximant from this manifold. This behavior will be called "eventual uniqueness". Theorem 3 indicates the sharpness of these two results.We emphasize that S N is not only a nonlinear manifold, but also a nonclosed subset of L 2 [0, 1]. Hence, arguments regarding existence, uniqueness, and characterization of best approximants are nontrivial. Since it has been shown in [1] that, for every positive integer TV, any continuous function has at least one best L 2 [0, 1] approximant from S N , we are only concerned with uniqueness and eventual uniqueness of best approximants in this paper.