Subdividing the Polar of a Face

Let S be a convex polytope. The set of all valid inequalities carries the structure of a convex polytope S, called the polar (polytope) of S. The facial structure of the polar provides information for each of its points: two points a and b are in the same face of S if and only if the faces of S obtained by intersecting it with the hyperplanes given respectively by a and b coincide. Suppose now that S is a face of another polyhedron P . Then the points of S carry some additional information: the set of faces of P which one can obtain by “rotating” the hyperplane given by a point. This additional information is captured by the structure of a polyhedral complex subdividing S . In this paper, we study this subdivision for the following examples: The Birkhoff polytope as a face of the matching polytope; the permutahedron as a facet of another permutahedron; the Symmetric Traveling Salesman Polytope, also known as Hamiltonian Cycle polytope, as a face of the connected Eulerian multi-graph polyhedron, also known as Graphical Traveling Salesman Polyhedron.