A notion of robustness and stability of manifolds


DENİZ A., Kocak S., RATİU A. V.

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, vol.342, no.1, pp.524-533, 2008 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 342 Issue: 1
  • Publication Date: 2008
  • Doi Number: 10.1016/j.jmaa.2007.12.025
  • Journal Name: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.524-533
  • Keywords: thickness, positive reach, stability of manifolds, Hausdortf distance
  • Anadolu University Affiliated: Yes

Abstract

Starting from the notion of thickness of Parks we define a notion of robustness for arbitrary subsets of R-k and we investigate its relationship with the notion of positive reach of Federer. We prove that if a set M is robust, then its boundary a M is of positive reach and conversely (under very mild restrictions) if partial derivative M is of positive reach, then M is robust. We then prove that a closed non-empty robust set in R-k (different from R-k) is a codimension zero submanifold of class C-1 with boundary. As a partial converse we show that any compact codimension zero submanifold with boundary of class C-2 is robust. Using the notion of robustness we prove a kind of stability theorem for codimension zero compact submanifolds with boundary: two such submanifolds, whose boundaries are close enough (in the sense of Hausdorff distance), are diffeomorphic. (C) 2007 Elsevier Inc. All rights reserved.