The Hirsch Conjecture (1957) stated that the graph of a d-dimensional polytope with n facets
cannot have (combinatorial) diameter greater than n–d. That is, any two vertices of the …

Abstract The Hirsch Conjecture, posed in 1957, stated that the graph of ad-dimensional
polytope or polyhedron with n facets cannot have diameter greater than n− d. The conjecture …

We often rely on censuses of triangulations to guide our intuition in $3 $-manifold topology.
However, this can lead to misplaced faith in conjectures if the smallest counterexamples are …

We investigate the diameter of a natural abstraction of the 1-skeleton of polyhedra. Although
this abstraction is simpler than other abstractions that were previously studied in the …

We show that nonlinear optimization techniques can successfully be applied to realize and
to inscribe matroid polytopes and simplicial spheres. In order to show non-realizability of …

From the point of view of optimization, a critical issue is relating the combinatorial diameter
of a polyhedron to its number of facets f and dimension d. In the seminal paper of Klee and …

This book introduces convex polytopes and their graphs, alongside the results and
methodologies required to study them. It guides the reader from the basics to current …

We introduce topological prismatoids, a combinatorial abstraction of the (geometric)
prismatoids recently introduced by the second author to construct counter-examples to the …

Neighborly polytopes are those that maximize the number of faces in each dimension
among all polytopes with the same number of vertices. Despite their extremal properties …

Let Δ (d, n) be the maximum possible diameter of the vertex-edge graph over all d-
dimensional polytopes defined by n inequalities. The Hirsch bound holds for particular n and …