ITERATED FIBER POLYTOPES

The construction of the fiber polytope Sigma(P, Q) of a projection pi: P --> Q of polytopes is extended to flags of projections. While the f aces of the fiber polytope are related to subdivisions of Q induced by the faces of P, those of an iterated fiber polytope are related to di screte homotopies between polyhedral subdivisions. In particular, in t he case of projections R(n+2) --> R(3) --> R(2) --> R(1), starting wit h an (n + 1)-simplex, vertices of the successive iterates correspond t o, respectively, subsets, permutations and sequences of permutations o f an n-set. The first iterate will always be combinatorially an n-cube , and, under certain conditions, the second will have the structure of the (n - 1)-dimensional permutohedron.