A Study of Some Geometric Structures on Inner Metric Spaces

Inner metric spaces, characterized by the approximate midpoint property, play an important role in metric geometry and in the analysis of length structures on metric spaces. This paper investigates structural properties of inner metric spaces, focusing on the relationship between the inner metric condition, approximate midpoints, and geodesicity. We first revisit the definition of inner metric spaces and establish that every inner metric space admits an approximate midpoint. We then show that, when an inner metric space is proper, it is geodesic. The arguments rely on the notions of length of curves and rectifiable curves to relate distance and curve length within this class of spaces. These results clarify how the inner metric property is linked to geodesic behavior and contribute to a deeper understanding of metric spaces that can be treated as length spaces.