We wish to draw attention to an interesting and promising interaction of two theories. On the one hand, it is the theory of pseudo-triangulations which was useful for implicit solution of the carpenter’s rule problem and proved later to give a nice tool for graph embeddings. On the other hand, it is the theory of hyperbolic virtual polytopes which arose from an old uniqueness conjecture for con...

We describe a provably complete algorithm for the generation of a tight, possibly exact, superset of all combinatorially distinct simple n-facet polytopes in Rd, along with their graphs, f -vectors, and face lattices. The technique applies repeated cutting planes and planar sweeps to a d−simplex. Our generator has implications for several outstanding problems in polytope theory, including conje...

Toric origami manifold is a generalization of a symplectic toric manifold: it is a manifold with torus action and a compatible 2-form degenerating in a nice, controllable way. In the same way as symplectic toric manifolds are classified by Delzant polytopes, symplectic toric manifolds are classified by certain collections of Delzant polytopes, called origami templates. We are interested in the ...

We relate the maximum semidefinite and linear extension complexity of a family of polytopes to the cardinality of this family and the minimum pairwise Hausdorff distance of its members. This result directly implies a known lower bound on the maximum semidefinite extension complexity of 0/1-polytopes. We further show how our result can be used to improve on the corresponding bounds known for pol...

In this paper, a thermodynamic analysis of Bouc-Wen models endowed with both strength and stiffness degradation is provided. It is based on the relationship between the flow rules of these models and those of the endochronic plasticity theory with damage, discussed in a companion paper (Erlicher and Point, 2007). Using the theoretical framework of that extended endochronic theory, it is shown t...

These lecture notes treat some current aspects of two closely interrelated topics from the theory of convex polytopes: the shapes of f -vectors, and extremal constructions. The study of f -vectors has had huge successes in the last forty years. The most fundamental one is undoubtedly the “g-theorem,” conjectured by McMullen in 1971 and proved by Billera & Lee and Stanley in 1980, which characte...