Continuities of Metric Projection and GeometricConsequences

✍ Scribed by Robert Huotari; Wu Li

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 357 KB
• Volume: 90
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1996.3018

✦ Synopsis

We discuss the geometric characterization of a subset K of a normed linear space via continuity conditions on the metric projection onto K. The geometric properties considered include convexity, tubularity, and polyhedral structure. The continuity conditions utilized include semicontinuity, generalized strong uniqueness and the non-triviality of the derived mapping. In finite-dimensional space with the uniform norm we show that convexity is equivalent to rotation-invariant almost convexity and we characterize those sets every rotation of which has continuous metric projection. We show that polyhedral structure underlies generalized strong uniqueness of the metric projection.

1997 Academic Press, Inc.

1. Introduction

Suppose (X, & } &) is a normed linear space and K is a closed subset of X. Let dist(x, K ) :=inf[&x& y& : y # K ] for x # X. The metric projection from X onto K is the set-valued mapping 6 K defined by