Research Article

Farthest point problem and partial statistical continuity in normed linear spaces

Published in: Quaestiones Mathematicae
Volume 45 , issue 4 , 2022 , pages: 595–604

DOI: 10.2989/16073606.2021.1886193

Author(s): Sudeshna Basu , India , Lakshmi Kanta Dey , India , Sumit Som , India

Keywords: 46B20 , 41A65 , 41A50 , 40A35 , 46B20 , 41A65 , 41A50 , 40A35

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.

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