An object representation scheme which allow_ fast Eucltdea,7 distance cal,:vlatu_rz is prese_ted. The object model eztends the polytope model by representLny objects as the convez hull of a finite set of spheres. An algorithm for calculating distances between objects is developed which is linear in the total number of spheres specifying the two objects.

We prove that the ranks of the subsets and the activities of the bases of a matroid define valuations for the subdivisions of a matroid polytope into smaller matroid polytopes.

We introduce new basis exchange axioms for matroids and oriented matroids. These new axioms are special cases of exchange properties for a more general class of combinatorial structures, Coxeter matroids. We refer to them as “properties” in the more general setting because they are not all equivalent, as they are for ordinary matroids, since the Symmetric Exchange Property is strictly stronger ...

Abstract. Let δ(P) be the δ-vector of a reflexive polytope P ⊂ R of dimension d and δ(P) the δ-vector of the dual polytope P ⊂ R. In general, δ(P) = δ(P) does not hold. In this paper, we give a higher-dimensional construction of a reflexive polytope whose δ-vector equals the δ-vector of the dual polytope. In particular, we consider the case that the reflexive polytope and the dual polytope are ...

In this paper we define a combinatorial object called a pedigree, and study the corresponding polytope, called the pedigree polytope. Pedigrees are in one-to-one correspondence with the Hamiltonian cycles on Kn. Interestingly, the pedigree polytope seems to differ from the standard tour polytope, Qn with respect to the complexity of testing whether two given vertices of the polytope are nonadja...

The shape of any facet of a 3-polytope can be preassigned. That is, given any 3-polytope P a facet F of P and a polygon F’ with the same number of sides as F, there is an isomorphic 3-polytope P’ such that F is a facet of P’ and the isomorphism takes F onto F (see [ 21). It is also known that given any simple circuit C in the graph of a 3-polytope P, there is an isomorphic 3-polytope P’ such th...

The classical polytope associated with the symmetric traveling salesman problem (STSP) is usually studied as embedded in the standard subtour elimination polytope. Several classes of facet defining inequalities of the STSP polytope are used in practical enumeration algorithms. In this paper we consider the basic objects called pedigrees which are in 1-1 correspondence with the tours. The convex...