On some special subspaces of a Banach space, from the perspective of best coapproximation (Q6582388)

Let \(X\) be a real Banach space. We say that \(X\) satisfies Kadets-Klee property if whenever \(\{x_{n}\}\) is a sequence in \(X\) and \(x \in X\) such that \(x_{n} \to x\) and \(||x_{n}|| \to ||x||\), it follows that \(x_{n} \to x\). The space \(X\) is called a polyhedral Banach space if each of its finite dimensional subspace is polytope. Given \(x,y \in X\), we say that \(x\) is Birkhoff-James orthogonal to \(y\), written as \(x \perp_{B} y\) , if \(||x+\lambda y|| \geq ||x||\), for all \(\lambda \in \mathbb{R}\). For a subspace \(Y\) of \(X\) and \(x \in X\), we say that \(y_{0} \in Y\) is a best coapproximation to \(x\) out of \(Y\) if \(||y_{0}-y|| \leq ||x-y||\) for all \(y \in Y\), equivalently, if \(Y \perp_{B} (x-y_{0}) \), i.e. \(y \perp_{B} (x-y_{0})\) for all \(y \in Y\).\N\NLet \( \epsilon \in [0,1)\). Then for \(x, y \in X\), \(x\) is said to be \(\epsilon\)-Birkhoff-James orthogonal to \(y\) written as \(x \perp_{B}^{\epsilon} y\), if \(||x+\lambda y|| \geq (1-\epsilon)||x||\), for each \(\lambda \in \mathbb{R}\). For a subspace \(Y\) of \(X\) and \(x \in X\), we say that \(y_{0} \in Y\) is an \(\epsilon\)-best coapproximation to \(x\) out of \(Y\) if \(Y \perp_{B}^{\epsilon} (x-y_{0})\). The subspace \(Y\) is said to be\N\N\begin{itemize}\N\item [(i)] anti-coproximinal if for any given \(x\in X\backslash Y\), there does not exist any best coapproximation to \(x\) out of \(Y\),\N\item[(ii)] strongly anti-coproximinal if for any given \(x\in X\backslash Y\) and for any \(\epsilon \in [0,1)\), there does not exist \(\epsilon\) best coapproximation to \(x\) out of \(Y\).\N\end{itemize}\N\NIn this paper, the authors study best coapproximation problem in Banach space using Birkhoff-James orthogonal techniques. They characterize the anti-coproximinal subspaces in a smooth Banach space and using that they characterize Hilbert space among smooth Banach spaces. They also present a necessary condition for a subspace to be strongly anti-coproximinal in a reflexive Banach space, whose dual space satisfies the Kadets-Klee property. A sufficient condition for the same is obtained in any Banach space. They characterize the strongly anti-coproximinal subspaces in a finite-dimensional polyhedral Banach space and show that the strongly anti-coproximinal subspaces of such spaces possess some nice geometrical structures. The anti-coproximinal and the strongly anti-coproximinal subspaces of two distinguished polyhedral Banach spaces \(\ell_{\infty}^{n}\) and \(\ell_{1}^{n}\) have also been characterized in this paper.