A Note on farthest point problem in banach spaces

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Som S., SAVAŞ E.

Miskolc Mathematical Notes, vol.20, no.2, pp.1237-1243, 2019 (SCI-Expanded, Scopus) identifier

  • Publication Type: Article / Article
  • Volume: 20 Issue: 2
  • Publication Date: 2019
  • Doi Number: 10.18514/mmn.2019.2834
  • Journal Name: Miskolc Mathematical Notes
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.1237-1243
  • Keywords: Farthest point map, Ideal, Partial ideal statistical continuity, Uniquely remotal
  • Open Archive Collection: AVESIS Open Access Collection
  • Uşak University Affiliated: Yes

Abstract

Farthest point problem states that "Must every uniquely remotal set in a Banach space be singleton?" In this paper we introduce the notion of partial ideal statistical continuity of a function which is way weaker than continuity of a function. We give an example to show that partial ideal statistical continuity is weaker than continuity. In this paper we use Ideal summability to give some answers to FPP problem which improves the result in [13]. We prove that if E is a non-empty, bounded, uniquely remotal subset in a real Banach space X such that E has a Chebyshev center c and the farthest point map F: X→E restricted to [c,F(c)] is partially ideal statistically continuous at c then E is singleton.