A note on farthest point problem in Banach spaces

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 W X ! E restricted to OEc; F .c/ is partially ideal statistically continuous at c then E is singleton.

A Note on Uniquely Remotal Sets in 2-Banach Spaces M. Gurdal and E. Savas

“…Let T be a non-empty, bounded subset of a real 2-Banach space E. If T is remotal such that T has Chebyshev center c and the extracted farthest point mapF : E → T restricted to [c, F (c)] is partially statistically continuous at c in (E, ∥•, •∥) , then T is singleton.From Theorem 2 of Som and Savaş[25] we have Theorem 2 Let E be a real 2-Banach space and T be a non-empty, bounded, uniquely remotal set admitting a Chebyshev center c. If T is not a singleton, then the farthest point map F , restricted to (c, F (c)) is not partially statistically continuous at c. Proof. The uniquely remotal set T has a Chebyshev center c.Let x ∈ (c, F (c)].…”