Joonsegment
Joonsegment is a concept in metric geometry and related fields that generalizes the idea of a line segment between two points in a space equipped with a distance function. In a metric space (X,d), the joonsegment J(a,b) between points a and b is defined as the union of all points that lie on at least one length-minimizing path from a to b. Equivalently, it is the set of points that belong to some shortest path joining a to b. In Euclidean space, where the shortest paths are straight lines, J(a,b) coincides with the ordinary line segment [a,b]. In spaces that admit multiple geodesics between a and b, J(a,b) can be larger, consisting of the points on all such geodesics.
History and terminology clarify that the term was introduced in the mathematical literature to study generalized
Computation and examples vary by setting. In graphs, J(a,b) is the subgraph formed by all vertices and
Applications include robotics path planning, computer graphics, geographic information systems, and the study of generalized convexity
See also: geodesic, line segment, shortest path, convex hull, geodesic convexity.