We describe and analyze a new construction that produces new Eulerian lattices from old ones. It specializes to a construction that produces new strongly regular cellular spheres (whose face lattices are Eulerian). The construction does not always specialize to convex polytopes; however, in a number of cases where we can realize it, it produces interesting classes of polytopes. Thus we produce ...

A renowned theorem of Blind and Mani, with a constructive proof by Kalai and an efficiency proof by Friedman, shows that the whole face lattice of a simple polytope can be determined from its graph. This is part of a broader story of reconstructing face lattices from partial information, first considered comprehensively in Grünbaum’s 1967 book. This survey paper includes varied results and open...

We introduce the wedge product of two polytopes. The wedge product is described in terms of inequality systems, in terms of vertex coordinates as well as purely combinatorially, from the corresponding data of its constituents. The wedge product construction can be described as an iterated “subdirect product” as introduced by McMullen (1976); it is dual to the “wreath product” construction of Jo...

Self testing of integrated circuits by random patterns has several technical and economical advantages. But there exists a large number of circuits which cannot be randomly tested, since the fault coverage achieved that way would be too low. In this paper we show that this problem can be solved by unequiprobable random patterns, and an efficient procedure is presented compu ting the specific op...

The Random-Facet algorithm is a randomized variant of the simplex method which is known to solve any linear program with n variables and m constraints using an expected number of pivot steps which is subexponential in both n and m. This is the theoretically fastest simplex algorithm known to date if m ≈ n; it provably beats most of the classical deterministic variants which require exp(Ω(n)) pi...

in conjunction with Euler’s formula, f1 = f0 + f2 − 2; see also [6, Sect. 10.3]. Moreover, the cone of flag vectors of 3-polytopes is also given by these two inequalities, together with the usual Dehn– Sommerville relations, such as f02 = 2f1 and f012 = 4f1. Here the extreme cases, polytopes whose f or flag vectors lie at the boundary of the cone, are given by the simplicial polytopes (for whic...