Metric constructions and fixed point theorems in product spaces

The paper studies a general scheme for constructing metrics on a product of metric spaces by means of a family of continuous convex functions. This construction includes the conventional $p$-metrics and generates metrics that are topologically equivalent to the conventional ones. As an application, we study fixed point and approximate fixed point properties for nonexpansive maps on a product space equipped with the constructed metric. We show that existing fixed point results of this type are consequences of our framework. Examples are provided to illustrate the established results. The construction machinery is also used to study products of length and geodesic spaces. The obtained results encompass existing ones and provide a background for potential studies of fixed point properties on these product spaces.