A study on best simultaneous approximation through Chebyshev centers and related geometric properties in Banach spaces

Teena, Thomas and Paul, Tanmoy (2023) A study on best simultaneous approximation through Chebyshev centers and related geometric properties in Banach spaces. PhD thesis, Indian Institute of Technology Hyderabad.

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Abstract

In this work, we study the existence of (restricted) Chebyshev centers and the set-valued generalization of strong proximinality in Banach spaces. We mainly explore the concepts above in L1-predual spaces and their subspaces. Although it is well-known that a Chebyshev center exists for compact subsets of an L1-predual space, we approach this problem differently. Interestingly, this approach leads us to an explicit description of the Chebyshev centers of the compact subsets of the spaces in question. Furthermore, we establish the validity of a geometric identity in terms of the (restricted) Chebyshev radius in L1-predual spaces and characterize L1-predual spaces using it. This identity was first established in 2000 by R. Esp´ınola, A. Wi´snicki and J. Wo´sko for the space of real-valued continuous functions on a compact Hausdorff space S, denoted by C(S), which forms a major subclass of the L1-predual spaces. We also yield a few geometric characterizations of the ideals in L1-predual spaces. In particular, we obtain characterizations for a compact convex subset of a locally convex topological space to be a Choquet simplex. The study of strong (ball) proximinality gained momentum in the recent years and the main motivation to study this property is it results in some “nice” continuity properties of the metric projections. With the same motivation, we extend the study of the set-valued generalization of strong proximinality, which was initiated by J. Mach in the literature. This generalization is termed as property-(P1) by Mach. For a non-empty closed convex subset V of a Banach space X and a family F of non-empty closed bounded subsets of X, property-(P1) is defined for a triplet (X, V, F). We study the interconnnection between property-(P1) of a subspace and that of its closed unit ball in a Banach space in detail. Expanding on some of the works by C. R. Jayanarayanan and S. Lalithambigai, we establish the equivalence of strong ball proximinality and property-(P1) of the closed unit ball of the finite co-dimensional subspaces of the L1-predual spaces. For a general subspace of a Banach space, we prove that property-(P1) of the closed unit ball of the subspace implies property-(P1) of the subspace itself. We also establish a similar implication in the case of the Hausdorff metric continuity of the restricted Chebyshev-center map of the subspace and that of its closed unit ball. We further investigate property-(P1) and the continuity properties of the restricted Chebyshev center maps in vector-valued continuous function spaces. We derive that if Y is a proximinal finite co-dimensional closed linear subspace of c0 then the closed unit ball of Y satisfies property-(P1) for the non-empty closed bounded subsets of ℓ∞ and the restricted Chebyshev-center map of the closed unit ball of Y is Hausdorff metric continuous on the class of non-empty closed bounded subsets of ℓ∞ with equi-bounded restricted Chebyshev radii. We also prove a few stability results of property-(P1) and the continuity of the restricted Chebyshev-center maps in ℓ∞-direct sum of Banach spaces. Finally, we discuss a few positive results on the existence of restricted Chebyshev centers and property-(P1) of an ideal in an L1-predual space and in particular, of an L1-predual space in its bidual.

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