Best Approximations on Quasi-Cone Metric Spaces
Ahmad Hisbu Zakiyudin, Kistosil Fahim, Nur Millatul Af-Idah, Felix Lyanto Setiawan, Ririana Annisatul Lathifah, Nur Izzah Nurdin, Tantri Tarisma
TL;DR
The paper tackles best-approximation problems in quasi-cone metric spaces, extending classical approximation theory to distances that take values in an ordered Banach space $B$. It defines forward and backward best approximations, introduces Chebyshev, quasi-Chebyshev, and pseudo-Chebyshev concepts, and proves necessary and sufficient criteria for inclusion in the best-approximation sets via an auxiliary map $f: Q -> B$. A central result, Theorem 1, provides a characterization for subsets of the best-approximation set, complemented by lemmas establishing conditions for membership. These results lay a foundational framework for optimization and analysis in generalized metric structures.
Abstract
This paper focuses on the best approximation in quasi-cone metric spaces, a combination of quasi-metrics and cone metrics, which generalizes the notion of distance by allowing it to take values in an ordered Banach space. We explore the fundamental properties of best approximations in this setting, such as the best approximation sets and the Chebyshev sets.