Metric Curvature, Folding, and Unique Best Approximation

Abstract

In his paper, the concepts of metric curvature and folding of a Cl-representable manifold in a normed linear space are studied. With certain restrictions on the metric curvature and/or folding, one can obtain a neighborhood of unique best approximation from the manifold, and in some cases, the manifold can be shown to be Chebyshev. Several familiar examples, including some classes of T-polynomials, are given.1. Introduction. The purpose of this paper is to study unique best approximation from subsets of normed linear spaces which are Cl-manifolds with boundary. In order to study such problems from a general geometric vantage point, John R. Rice [12] introduced the concepts of folding and metric curvature (originally called curvature) in smooth, rotund, and finite-dimensional spaces. These concepts were generalized to uniformly smooth spaces by two of the authors in [13] (see also [5], [6]). In 3 and 4 of the present work, we closely examine these concepts for Cl-manifolds with boundaries and obtain results which demonstrate several connections between local uniqueness, metric curvature, and folding.In C[a, b], there are nonlinear sets, for instance, the set of rational functions of degree no greater than (m, n), that are Chebyshev. We show that the fact that Haar embedded manifolds are Chebyshev (proved in a special case by Daniel Wulbert 17] and generalized by D. Braess [2]) follows from general results on the metric curvature of the manifold (see Theorem 6.1). For example, we see in 7 that the set N is Chebyshev in C[a, b], where 0 < a < b.In L([a, b], x) it is well known [14, p. 368] that a nonconvex boundedly compact subset is not Chebyshev (i.e., there is a point which does not have a unique best approximation from the set). In 7 we exhibit for the first time some familiar subsets M of L2 ([a, b],/x) each of which is a Cl-manifold with boundary and has a neighborhood of unique best approximation for M. Thus, for points in this neighborhood of M, steepest descent methods may be attempted. Many nonlinear regression problems fall into the above category.Both of the above results are special cases of Theorem 5.1 in this paper, which basically states that every manifold M with boundary which has finite metric curvature and positive folding has a neighborhood of unique best approximation from M. * Received by the editors August 16, 1974.

On the smoothness of best 𝐿₂ approximants from nonlinear spline manifolds

“…We first introduce some notation which will expedite the presentation. The notation is the same as found in [3]. X will denote a normed linear space and A a subset of X.…”

“…of [3] which states that a(m) < °° for all m E M implies that PM: X\M -► M is a surjection. The proof of this theorem is essentially the same as that of Theorem 5.1 in [3] and will be omitted.…”

“…of [3] which states that a(m) < °° for all m E M implies that PM: X\M -► M is a surjection. The proof of this theorem is essentially the same as that of Theorem 5.1 in [3] and will be omitted. As in the proof of Theorem 5.1, one proceeds by contradiction and finally contradicts the bound of a(m) using the Nonzero Index Theorem.…”

“…If M is a C1 -representable manifold for which the kernel of g'(6) is trivial then it was shown in [3] that fld(g'(d)) =£ 0. Thus, we obtain the analog of Theorem 5.2…”

The minimum norm projection on 𝐶²-manifolds in 𝑅ⁿ

“…R. Rice has studied existence of the map PM in terms of the radius of curvature and established continuity of PM on general grounds. In [1], [2], [4] and [5], we have an investigation of the existence of PM in a Banach space setting. The results use the notion of curvature.…”