Convex Polytopes : Examples and Conjectures by Günter M
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چکیده
منابع مشابه
On Kalai's Conjectures Concerning Centrally Symmetric Polytopes
In 1989 Kalai stated the three conjectures A, B, C of increasing strength concerning face numbers of centrally symmetric convex polytopes. The weakest conjecture,A, became known as the “3-conjecture”. It is well-known that the three conjectures hold in dimensions d ≤ 3. We show that in dimension 4 only conjectures A and B are valid, while conjecture C fails. Furthermore, we show that both conje...
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Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their importance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry to linear and combinatorial optimization. In this chapter we try to give a short introduction, provide a sketch of ...
متن کاملBasic Properties of Convex Polytopes
Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their importance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry (toric varieties) to linear and combinatorial optimization. In this chapter we try to give a short introduction, pr...
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These lecture notes treat some current aspects of two closely interrelated topics from the theory of convex polytopes: the shapes of f -vectors, and extremal constructions. The study of f -vectors has had huge successes in the last forty years. The most fundamental one is undoubtedly the “g-theorem,” conjectured by McMullen in 1971 and proved by Billera & Lee and Stanley in 1980, which characte...
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The edge polytope of a finite graph G is the convex hull of the columns of its vertex-edge incidence matrix. We study extremal problems for this class of polytopes. For k ∈ {2, 3, 5}, we determine the maximal number of vertices of kneighborly edge polytopes up to a sublinear term. We also construct a family of edge polytopes with exponentially-many facets.
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