# Convex Polytopes

@inproceedings{Grunbaum1967ConvexP, title={Convex Polytopes}, author={Branko Grunbaum and Geoffrey C. Shephard}, year={1967} }

Graphs, Graphs and Realizations Before proceeding to the graph version of Euler’s formula, some notation will be introduced. A (finite) abstract graph G consists of two sets, the set of vertices V = V(G) = {1, . . . , v} and the set of edges E = E(G) which consists of two-element subsets of V. If {u, w} is an edge, we write also uw for it and call w a neighbour of u and vice versa. If a vertex is contained in an edge or an edge contains a vertex then the vertex and the edge are incident. If two…

## 1,417 Citations

Eulerian edge refinements, geodesics, billiards and sphere coloring

- MathematicsArXiv
- 2018

This work constructs some ergodic billiards in 2-balls, where the geodesics bouncing off at the boundary symmetrically and which visit every interior edge exactly once, and tells that every 2-ball can be edge refined using interior edges to become Eulerian if and only if its boundary has length divisible by 3.

The Graph of the Hypersimplex

- Mathematics
- 2008

The (k,d)-hypersimplex is a (d-1)-dimensional polytope whose vertices are the (0,1)-vectors that sum to k. When k=1, we get a simplex whose graph is the complete graph with d vertices. Here we show…

BARYCENTRIC CHARACTERISTIC NUMBERS

- Mathematics
- 2015

If G is the category of finite simple graphs G = (V,E), the linear space V of valuations on G has a basis given by the f-numbers vk(G) counting complete subgraphs Kk+1 in G. The barycentric…

A Note on the Matching Polytope of a Graph

- MathematicsTEMA (São Carlos)
- 2019

The matching polytope of a graph G, denoted by M(G), is the convex hull of the set of the incidence vectors of the matchings G. The graph G(M(G)), whose vertices and edges are the vertices and edges…

On Fredholm determinants in topology

- MathematicsArXiv
- 2016

It is proved that the Fredholm characteristic det(1+A) takes values in {-1,1} and is equal to the Fermi characteristic, which is the product of the w(x), where w( x)=(-1)^dim(x).

SOME HAMILTONIAN COUNTEREXAMPLES

- Mathematics
- 1979

In this article we survey some interesting examples and counterexamples concerning hamiltonian graphs and related concepts. The discussion is divided into three areas: necessary or sufficient…

Orthogeodesic Point Set Embeddings of Outerplanar Graphs

- Mathematics
- 2015

Let G be a simple undirected n-vertex graph and let P ⊂ R be a general point set of size m, that is, a set of points where each two points have distinct xand distinct y-coordinates. We ask whether an…

On the vertex degrees of the skeleton of the matching polytope of a graph

- Mathematics
- 2017

The convex hull of the set of the incidence vectors of the matchings of a graph G is the matching polytope of the graph, M(G). The graph whose vertices and edges are the vertices and edges of M(G) is…

## References

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In a finite-dimensional EUCLIDean space E , a convex polytope P is a set which is the convex hull of a finite set; here such a set P will simply be called a polytope, or, if it is d-dimensional, a…

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- 1964

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- MathematicsCanadian Journal of Mathematics
- 1967

Let P be a d-polytope (that is, a d-dimensional convex polytope in Euclidean space) and for 0 ≤ j ≤ d – 1 let (i = 1, . . . ,ƒj(P)) represent its j-faces. Associated with each face is a non-negative…

Polytopes, Valuations, and the Euler Relation

- MathematicsCanadian Journal of Mathematics
- 1968

By a d-polytope we shall mean a d-dimensional convex polytope. We shall denote a j-dimensional face (or j-face) of a polytope by Fj . Every d-polytope P has proper j-faces for 0 ≦j ≦d — 1 and we…

Diagrams for centrally symmetric polytopes

- Mathematics
- 1968

In a paper [1] published in 1956, David Gale introduced the idea of representing a convex polytope by a diagram (now called a Gale diagram). Later work by Gale, T. S. Motzkin, and more recently by M.…

ON THE ENUMERATION OF CONVEX POLYTOPES AND COMBINATORIAL SPHERES.

- Mathematics
- 1969

Abstract : Combinatorial n-spheres and simplicial complexes are equivalent by stellar subdivisions to the boundary of the (n+1) -simplex. Best known examples are the boundary complexes of simplicial…