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...

A periodic graph in dimension d is a directed graph with a free action of Z with only finitely many orbits. It can conveniently be represented in terms of an associated finite graph with weights in Z. Here we use the weight sums along cycles in this associated graph to construct a certain polytope in R, which we regard as a geometrical invariant associated to the periodic graph. It is the unit ...

In this paper we consider the problem of efficiently constructing geodesic t-spanners. We consider finding spanners on the surface of a 3 dimensional polyhedron. If Steiner vertices are allowed on the surface of the polyhedron, then we are able to construct sparse t-spanners. If no Steiner vertices are allowed, then we establish lower bounds on the maximum node degree, depending on the spanning...

The operational Chow cohomology classes of a complete toric variety are identified with certain functions, called Minkowski weights, on the corresponding fan. The natural product of Chow cohomology classes makes the Minkowski weights into a commutative ring; the product is computed by a displacement in the lattice, which corresponds to a deformation in the toric variety. We show that, with rati...

We define a set of PBW-semistandard tableaux that is in weight preserving bijection with the monomials corresponding to integral points Feigin-Fourier-Littelmann-Vinberg polytope for highest modules symplectic Lie algebra. then show these parametrize bases homogeneous coordinate rings complete original and PBW degenerate flag varieties. From this construction, we provide explicit relations gene...