For a d-dimensional convex lattice polytope P , a formula for the boundary volume vol(∂P ) is derived in terms of the number of boundary lattice points on the first bd/2c dilations of P . As an application we give a necessary and sufficient condition for a polytope to be reflexive, and derive formulae for the f -vector of a smooth polytope in dimensions 3, 4, and 5. We also give applications to...

The space of torus translations and degenerations a projective toric variety forms associated to the secondary fan integer points in polytope corresponding variety. This is used identify moduli real with polytope. A configuration $${{\mathcal {A}}}$$ vectors gives an irrational simplex. We . For this, we develop theory varieties arbitrary fans. When rational, nonnegative part classical normal p...

The Birkhoff polytope is defined to be the convex hull of permutation matrices, Pσ ∀σ ∈ Sn. We define a second-order permutation matrix P [2] σ in R ×n corresponding to a permutation σ as (P [2] σ )ij,kl = (Pσ)ij(Pσ)kl. We call the convex hull of the second-order permutation matrices, the second-order Birkhoff polytope and denote it by B. It can be seen that B is isomorphic to the QAP-polytope,...

We use tropical geometry to compute the multidegree and Newton polytope of the hypersurface of a statistical model with two hidden and four observed binary random variables, solving an open question stated by Drton, Sturmfels and Sullivant in [6, Problem 7.7]. The model is obtained from the undirected graphical model of the complete bipartite graph K2,4 by marginalizing two of the six binary ra...

The secondary polytope of a point configuration A is a polytope whose face poset is isomorphic to the poset of all regular subdivisions ofA. While the vertices of the secondary polytope – corresponding to the triangulations ofA – are very well studied, there is not much known about the facets of the secondary polytope. The splits of a polytope, subdivisions with exactly two maximal faces, are t...

We express the matroid polytope PM of a matroid M as a signed Minkowski sum of simplices, and obtain a formula for the volume of PM . This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian Grk,n. We then derive analogous results for the independent set polytope and the associated flag matroid polytope of M . Our proofs are based on a natural...

We investigate a polytope (the QAP-Polytope) beyond a \natural" integer programming formulation of the Quadratic Assignment Problem (QAP) that has been used successfully in order to compute good lower bounds for the QAP in the very recent years. We present basic structural properties of the QAP-Polytope, partially independently also obtained by Rijal (1995). The main original contribution of th...

In a two-capacitated spanning tree of a complete graph with a distinguished root vertex v, every component of the induced subgraph on V\{v} has at most two vertices. We give a complete, non-redundant characterization of the polytope defined by the convex hull of the incidence vectors of two-capacitated spanning trees. This polytope is the intersection of the spanning tree polytope on the given ...

The edge formulation of the stable set problem is defined by two-variable constraints, one for each edge of a graph G, expressing the simple condition that two adjacent nodes cannot belong to a stable set. We study the fractional stable set polytope, i.e. the polytope defined by the linear relaxation of the edge formulation. Even if this polytope is a weak approximation of the stable set polyto...

We present a novel technique for encoding and decoding constant weight binary codes that uses a geometric interpretation of the codebook. Our technique is based on embedding the codebook in a Euclidean space of dimension equal to the weight of the code. The encoder and decoder mappings are then interpreted as a bijection between a certain hyper-rectangle and a polytope in this Euclidean space. ...