Farthest point problem and partial statistical continuity in normed linear spaces

Abstract

In this paper, we prove that if E is an uniquely remotal subset of a real normed linear space such that E has a Chebyshev center c ∈ and the farthest point map F : → E restricted to [c, F (c)] is partially statistically continuous at c, then E is a singleton. We obtain a necessary and sufficient condition on uniquely remotal subsets of uniformly rotund Banach spaces to be a singleton. Moreover, we show that there exists a remotal set M having a Chebyshev center c such that the farthest point map F : ℝ → M is not continuous at c but is partially statistically continuous there in the multivalued sense.