# An Introduction to Convex Polytopes

@inproceedings{Brndsted1982AnIT, title={An Introduction to Convex Polytopes}, author={Arne Br{\o}ndsted}, year={1982} }

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…

## 619 Citations

Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets

- Mathematics
- 1985

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…

Convex Polytopes and Unimodular Triangulations

- Mathematics
- 2018

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

- Mathematics
- 2011

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 (…

AC NU s

- Mathematics
- 1988

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

- Computer Science, MathematicsDiscret. Comput. Geom.
- 1998

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.

Minimax Equalities by Reconstruction of Polytopes

- Mathematics
- 2000

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

- MathematicsAnnals of Combinatorics
- 2019

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.…

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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

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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…

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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…

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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…

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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…

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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…