Farthest Point Problem and Partial Statistical Continuity in Normed Linear Spaces

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.

Characterization of $M$-compact sets via statistically convergent sequences

“…It was proved in [15] that a maximizing sequence is statistically maximizing. But the converse is not true.…”

“…(2) It was proved in [15] that if M ⊆ X. If {x n } n∈N is a statistically maximizing sequence in M then {x n } n∈N is a statistically maximizing sequence in M .…”

“…In this work, we prove the stability of M -compact sets with respect to ℓ p sum for 1 ≤ p < ∞. We also give a characterization of M -compact sets in terms of statistically maximizing sequence in a normed linear space introduced in [15]. We prove the stability of M -compact sets under ℓ p sum.…”