Abstract

We discuss a set-valued generalization of strong proximinality in Banach spaces, introduced by J. Mach [Continuity properties of Chebyshev centers, J. Approx. Theory 29/3 (1980) 223–230] as property-(P1)(P_1). For a Banach space XX, a closed convex subset VV of XX and a subclass F\mathcal{F} of the closed bounded subsets of XX, this property, defined for the triplet (X,V,F)(X,V,\mathcal{F}), describes simultaneous strong proximinality of VV at each of the sets in F\mathcal{F}. We establish that if the closed unit ball of a closed subspace of a Banach space XX possesses property-(P1)(P_1) for each of the classes of closed bounded, compact and finite subsets of XX, then so does the subspace. It is also proved that the closed unit ball of an MM-ideal in an L1L_{1}-predual space satisfies property-(P1)(P_{1}) for the compact subsets of the space. For a Choquet simplex KK, we provide a sufficient condition for the closed unit ball of a finite co-dimensional closed subspace of A(K)A(K) to satisfy property-(P1)(P_{1}) for the compact subsets of A(K)A(K). This condition also helps to establish the equivalence of strong proximinality of the closed unit ball of a finite co-dimensional subspace of A(K)A(K) and property-(P1)(P_1) of the closed unit ball of the subspace for the compact subsets of A(K)A(K). Further, for a compact Hausdorff space SS, a characterization is provided for a strongly proximinal finite co-dimensional closed subspace of C(S)C(S) in terms of property-(P1)(P_{1}) of the subspace and that of its closed unit ball for the compact subsets of C(S)C(S). We generalize this characterization for a strongly proximinal finite co-dimensional closed subspace of an L1L_{1}-predual space. As a consequence, we prove that such a subspace is a finite intersection of hyperplanes such that the closed unit ball of each of these hyperplanes satisfy property-(P1)(P_1) for the compact subsets of the L1L_1-predual space and vice versa. We conclude this article by providing an example of a closed subspace of a non-reflexive Banach space which satisfies 1121 \frac{1}{2}-ball property and does not admit restricted Chebyshev center for a closed bounded subset of the Banach space.

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Keywords

  • Property-(P1)
  • strong proximinality
  • restricted Chebyshev center
  • L1-predual
  • M-ideal
  • 112-ball property

Mathematics Subject Classification

41A65, 41A50; 52A07, 46E15

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Suggested citation

T. Thomas. “On Property-(P_1) in Banach Spaces.” Journal of Convex Analysis 29 (2022), No. 4, 975–994. https://doi.org/10.68381/jca29055

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