Smoothness of certain metric projections on Hilbert space
Smoothness of certain metric projections on Hilbert space
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DOI：
10.1090/s0002-9947-1973-0326252-2
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发表时间：
1973-10
影响因子：
1.3
通讯作者：
R. Holmes
中科院分区：
文献类型：
--
作者：
R. Holmes
研究了由Hubert空间的闭凸集定义的距离函数和度量投影的微分性质。前映射也被认为是在更一般的Banach空间的背景下。导论.本文研究了Hubert空间X中闭凸集K的最佳逼近问题。因为任何这样的集合都是切比雪夫集合，所以在X上定义了一个映射Pn：X -» K，它将X的每个元素分配给它在K中的最佳逼近（最近点）。映射PK通常称为X在K上的度量投影。众所周知，P n是X上的非扩张映射。在本工作中，我们将关注的微分性质的PK，在适当的光滑性假设的边界K。通过将这样一个简单的集合考虑为线段，很明显PK在X中的某些点上可能不具有双侧方向导数。更令人惊讶的是，J. Kruskal [5]最近构造了R3的一个凸子集，其相关的度量投影在R的某些点上没有单侧方向导数。另一方面，由于上述P n的Lipschitz连续性，Rademacher和Stepanoff的经典定理[3，p.216]保证了当X是有限维时，PK几乎处处（Frechet）可微。(This顺便说一句，观察结果对Kruskal在[5]第697页底部的问题给出了肯定的答案-）然而，当X是无限维的时候，不能保证Lipschitzian映射是可微的（参见。[8] 91-92]）。下面要给出的主要结果是（粗略地），如果对于某个x ∈ X，K的边界在P（U）附近属于Cp + 1类，则P n在P（K x）处垂直于K的射线的邻域上（因此特别是在x的邻域上）属于Cp类。证明最终归结为隐函数定理的应用。我们进一步获得了微分DPAx）的各种性质，并在1972年3月1日和1972年10月30日的修订版中被编辑收到。AMS（MOS）主题分类（1970年）。小学41 A65;中学46 B 99、46 C10。
A study is made of differential properties of the distance function and the metric projection defined by a closed convex subset of Hubert space. The former mapping is also considered within the context of more general Banach spaces. Introduction. This paper contains a contribution to the study of best approximation out of a closed convex set K belonging to a Hubert space X. As any such set is a Chebyshev set, there is defined on X a map P„: X —» K which assigns to each element of X its best approximation (nearest point) in K. The map PK is generally called the metric projection of X on K. P„ is well known to be a nonexpansive map on X. In the present work we shall be concerned with differential properties of PK, under appropriate smoothness assumptions about the boundary of K. By considering such a simple set as a line segment, it is clear that PK may fail to possess a two-sided directional derivative at certain points in X. More surprisingly, J. Kruskal [5] recently constructed a convex subset of R3 whose associated metric projection failed to have a one-sided directional derivative at some points of R . On the other hand, due to the aforementioned Lipschitz continuity of P„, the classical theorem of Rademacher and Stepanoff [3, p. 216] guarantees that PK is almost everywhere (Frechet) differentiable when X is finite dimensional. (This observation, incidentally, yields an affirmative answer to the question of Kruskal at the bottom of p. 697 of [5]-) However, when X is infinite dimensional, there is no guarantee that Lipschitzian maps are differentiable (cf. [8, pp. 91-92]). The main result to be presented below is (roughly) that if for some x e X the boundary of K is of class Cp + 1 near P^U), then P„ is of class Cp on a neighborhood of the ray normal to K at PKix) (hence in particular on a neighborhood of x). The proof ultimately reduces to an application of the implicit function theorem. We further obtain various properties of the differential DPAx), and in parReceived by the editors March 1, 1972 and, in revised form, October 30, 1972. AMS (MOS) subject classifications (1970). Primary 41A65; Secondary 46B99, 46C10.