Fiber polytopes (Q1197342)

The authors consider projections \(\pi: P\to Q\) between convex polytopes. For each \(x\in Q\) the fiber \(\pi^{-1}(x)\) is again a convex polytope, and the average of \(\pi^{-1}(x)\) over all \(x\in Q\) is called the fiber polytope \(\Sigma(P,Q)\). Its precise definition is given in terms of the Minkowski integral which is the average of the integral over all sections of \(\pi\). It is shown that the fiber polytope can be expressed as the Minkowski sum of finitely many fibers. Throughout the paper, various interesting properties of fiber polytopes are studied, in particular an interpretation of the combinatorial structure of \(\Sigma(P,Q)\) in terms of coherent subdivisions of \(Q\), relations between the boundary complexes of \(P\) and \(Q\), special cases like zonotopes as projections of cubes and centrally symmetric polytopes as projections of cross polytopes.