An Introduction to Convex Polytopes

  title={An Introduction to Convex Polytopes},
  author={Arne Br{\o}ndsted},
1 Convex Sets.- 1. The Affine Structure of ?d.- 2. Convex Sets.- 3. The Relative Interior of a Convex Set.- 4. Supporting Hyperplanes and Halfspaces.- 5. The Facial Structure of a Closed Convex Set.- 6. Polarity.- 2 Convex Polytopes.- 7. Polytopes.- 8. Polyhedral Sets.- 9. Polarity of Polytopes and Polyhedral Sets.- 10. Equivalence and Duality of Polytopes.- 11. Vertex-Figures.- 12. Simple and Simplicial Polytopes.- 13. Cyclic Polytopes.- 14. Neighbourly Polytopes.- 15. The Graph of a Polytope… 
Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets
This paper generalizes the Dehn-Sommerville equations for simplicial spheres to related classes of objects. The underlying motivation is to understand the combinatorial structure of arbitrary
A new basis of polytopes
  • G. Kalai
  • Mathematics
    J. Comb. Theory, Ser. A
  • 1988
Convex Polytopes and Unimodular Triangulations
The triangulation of a convex polytope is one of the most important topics in the classical theory of convex polytopes. In this chapter the modern treatment of triangulations of convex polytopes is
Minimizing the mass of the codimension-two skeleton of a convex, volume-one polyhedral region
Minimizing the mass of the codimension-two skeleton of a convex, volume-one polyhedral region by Ryan Christopher Scott In this paper we establish the existence and partial regularity of a (
A completely unimodal numbering of the m vertices of a simple d-dimensional polytope is a numbering 0, 1, . . . , m - 1 of the vertices such that on every k-dimensional face (2s ksd) there is exactly
Primal—Dual Methods for Vertex and Facet Enumeration
The known polynomially solvable classes of polytopes are extended by looking at the dual problems by proposing a new class of algorithms that take advantage of this phenomenon.
Completely unimodal numberings of a simple polytope
  • K. Hoke
  • Mathematics
    Discret. Appl. Math.
  • 1988
Minimax Equalities by Reconstruction of Polytopes
Given a quasi-concave-convex function f : X × Y → R defined on the product of two convex sets we would like to know if infY supX f = supX infY f . In [4] we showed that that question is very closely
The Numbers of Edges of 5-Polytopes with a Given Number of Vertices
A basic combinatorial invariant of a convex polytope P is its f-vector $$f(P)=(f_0,f_1,\dots ,f_{\dim P-1})$$f(P)=(f0,f1,⋯,fdimP-1), where $$f_i$$fi is the number of i-dimensional faces of P.


A proof of the lower bound conjecture for convex polytopes.
1* Definitions and preliminary results* If v is a vertex of a d-polytope P then the antistar of v in P, denoted ast(v, P), is the set of all ά-faces of P that miss v, 0 <Ξ k ̂ d — 1. If H is a
Polytope pairs and their relationship to linear programming
Abstract : An important development in the theory of (convex) polytopes was the determination by Barnette and McMullen of the minimum and maximum of v(P) (number of vertices of P) as P ranges over
Sufficiency of McMullen's conditions for $f$-vectors of simplicial polytopes
For convex d-polytope P let ft{P) equal the number of faces of P of dimension i, 0 < i < d 1. f(P) = (f0(P)9 . . . , fd^QP)) is called the f vector of P An important combinatorial problem is the
The unimodality conjecture for convex polytopes
for some k, 0 < k < n. Empirical evidence has suggested that some important numerical sequences in combinatorics are unimodal. For instance, unimodality conjectures have been made for the
The minimum number of vertices of a simple polytope
Ad-polytope is ad-dimensional set that is the convex hull of a finite number of points. Ad-polytope is simple provided each vertex meets exactlyd edges. It has been conjectured that for simple
The maximum numbers of faces of a convex polytope
In this paper we give a proof of the long-standing Upper-bound Conjecture for convex polytopes, which states that, for 1 ≤ j d v , the maximum possible number of j -faces of a d -polytope with v
The Minimum Number of Faces of a Simple Polyhedron
The numbers of faces of simplicial polytopes
In this paper is considered the problem of determining the possiblef-vectors of simplicial polytopes. A conjecture is made about the form of the sclution to this problem; it is proved in the case