Deformed Products and Maximal Shadows of Polytopes

We present a construction of deformed products of polytopes that has as special cases all the known constructions of linear programs with \many pivots," starting with the famous Klee-Minty cubes from 1972. Thus we obtain sharp estimates for the following geometric quantities for d-dimensional simple polytopes with at most n facets: the maximal number of vertices on an increasing path, the maximal number of vertices on a \greedy" greatest increase path, and the maximal number of vertices of a 2-dimensional projection. This, equivalently, provides good estimates for the worst-case behaviour of the simplex algorithm on linear programs with these parameters with the worst-possible, the greatest increase, and the shadow vertex pivot rules. The bounds on the maximal number of vertices on an increasing path or a greatest increase path unify and slightly improve a number of known results. One bound on the maximal number of vertices of a 2-dimensional projection is new: we show that a 2-dimensional projection of a d-dimensional polytope with n facets may have as many as (n bd=2c) vertices for xed d. This provides the same bound for the worst-case behaviour of the simplex algorithm with the shadow vertex pivot rule. The maximal complexity of shadows in xed dimension is also relevant for problems of Computational Geometry. We give a new algorithm for the construction of the shadow of a d-dimensional polytope. However, we nd that for even d 4 the polars of cyclic polytopes, C d (n) , which have the maximal number of vertices for any given n, do not maximize the shadow: for example, any 2-dimensional projection of C 4 (n) has not more than 3n vertices.