The authors consider a generic configuration of regions, consisting of a collection of distinct compact regions $\{ \Omega_i\}$ in $\mathbb{R}^{n+1}$ which may be either regions with smooth boundaries disjoint from the others or regions which meet on their piecewise smooth boundaries $\mathcal{B}_i$ in a generic way. They introduce a skeletal linking structure for the collection of regions which simultaneously captures the regions' individual shapes and geometric properties as well as the ``positional geometry'' of the collection. The linking structure extends in a minimal way the individual ``skeletal structures'' on each of the regions. This allows the authors to significantly extend the mathematical methods introduced for single regions to the configuration of regions.

Table of Contents:
Introduction Part 1. Medial/Skeletal Linking Structures: Multi-region configurations in $\mathbb{R}^{n+1}$ Skeletal linking structures for multi-region configurations in ${\mathbb R}^{n+1}$ Blum medial linking structure for a generic multi-region configuration Retracting the full Blum medial structure to a skeletal linking structure Part 2. Positional Geometry of Linking Structures: Questions involving positional geometry of a multi-region configuration Shape operators and radial flow for a skeletal structure Linking flow and curvature conditions Properties of regions defined using the linking flow Global geometry via medial and skeletal linking integrals Positional geometric properties of multi-region configurations Part 3. Generic Properties of Linking Structures via Transversality Theorems: Multi-distance and height-distance functions and partial multi-jet spaces Generic Blum linking properties via transversality theorems Generic properties of Blum linking structures Concluding generic properties of Blum linking structures Part 4. Proofs and Calculations for the Transversality Theorems: Reductions of the proofs of the transversality theorems Families of perturbations and their infinitesimal properties Completing the proofs of the transversality theorems Appendix A. List of frequently used notation Bibliography.