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 X such that E has a Chebyshev center c Є X and the farthest point map F : X → 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.