The Blum medial axis of a single region in R^{n+1} with smooth boundary is a skeleton-like topological structure that captures shape and geometric properties of the region and its boundary. We introduce a structure, called the Blum medial linking structure, which extends the advantages of the medial axis to configurations of multiple regions in order to capture both their individual and "positional" or relative geometry. We use singularity theory to classify the local normal forms of medial linking for generic configurations of regions in dimensions n < or = 6, and show how invariants of the geometry of the regions and their complement may be computed directly from the linking structure. We conclude with applications of the linking structure to the analysis of multiple objects in medical images.