A new method to construct the tight span of a finite subset of the Manhattan plane via trimming

In this study a new algorithm is given to construct the tight span of a finite subset of the Manhattan plane. Initially, with the help of an equivalence relation defined on a metric space, the so-called trimming space is obtained and with successive iterations a pseudometric graph called trimming cylinder is constructed. As an application of these definitions, a different proof is given for the property: If a metric on a set satisfies the four-point condition, then the metric space can be embedded into a weighted tree. Secondly, the metric center for a point in the Manhattan plane is defined and the coordinates for the metric center is specifically determined. Using the related lemmas for metric center, the trimming cylinder of a finite metric space is embedded isometrically into the Manhattan plane. Finally, the tight span of the trim metric space which is obtained from the initial given metric space is constructed by the given algorithm.