Tight Polyhedral Submanifolds and Tight Triangulations

This volume is an introduction and a monograph about tight polyhedra. The treatment of the 2-dimensional case is self-contained and fairly elementary. It should also be useful for undergraduate seminars. Particular emphasis is given to the interplay of varios special disciplines, such as geometry, elementary topology, combinatorics and convex polytopes. A typical result relates tight submanifolds to combinatorial properties of their convex hulls. The chapters on higher dimensions generalize the 2-dimensional case using concepts from combinatorics and topology, such as combinatorial Morse theory. A number of open problems are also discussed.