Farthest Point Problem and Partial Statistical Continuity in Normed Linear Spaces

2020/05/27 by Som, Sumit, Dey, Lakshmi Kanta, Basu, Sudeshna · 1 citation
#46B20 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2005.13355

Abstract

In this paper, we prove that if E is a uniquely remotal subset of a real normed linear space X such that E has a Chebyshev center c ∈ 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 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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• A note on remotal and uniquely remotal sets in normed linear spaces

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