On the best coapproximation problem in $\ell_1^n$

TL;DR

This paper investigates the best coapproximation problem in finite-dimensional  spaces using Birkhoff-James orthogonality, providing a complete characterization, an algorithmic approach, and identifying special subspaces.

Contribution

It offers a complete characterization of elements with best coapproximations in ^n, introduces an effective computational method, and identifies coproximinal and co-Chebyshev subspaces.

Findings

Characterization of elements with best coapproximations in ^n

Development of an algorithmic approach for coapproximation

Identification of coproximinal and co-Chebyshev subspaces

Abstract

We study the best coapproximation problem in the Banach space $ \ell_1^n, $ by using Birkhoff-James orthogonality techniques. Given a subspace $\mathbb{{Y}}$ of $\ell_1^n$, we completely identify the elements $x$ in $\ell_1^n,$ for which best coapproximations to $x$ out of $\mathbb{{Y}}$ exist. The methods developed in this article are computationally effective and it allows us to present an algorithmic approach to the concerned problem. We also identify the coproximinal subspaces and co-Chebyshev subspaces of $\ell_1^n$.