Some Aspects of the Combinatorial Theory of Convex Polytopes
نویسنده
چکیده
We start with a theorem of Perles on the k-skeleton, Skel k (P) (faces of dimension k) of d-polytopes P with d+b vertices for large d. The theorem says that for xed b and d, if d is suuciently large, then Skel k (P) is the k-skeleton of a pyramid over a (d ? 1)-dimensional polytope. Therefore the number of combinatorially distinct k-skeleta of d-polytopes with d + b vertices is bounded by a function of k and b alone. Next we replace b (the number of vertices minus the dimension) by related but deeper invariants of P , the g-numbers. For a d-polytope P there are d=2] invariants g 1 (P); g 2 (P); :::; g d=2] (P) which are of great importance in the combinatorial theory of polytopes. We study polytopes for which g k is small and carried away to related and slightly related problems.
منابع مشابه
Linear Programming, the Simplex Algorithm and Simple Polytopes
In the first part of the paper we survey some far reaching applications of the basis facts of linear programming to the combinatorial theory of simple polytopes. In the second part we discuss some recent developments concurring the simplex algorithm. We describe sub-exponential randomized pivot roles and upper bounds on the diameter of graphs of polytopes.
متن کاملNormal polytopes
In Section 1 we overview combinatorial results on normal polytopes, old and new. These polytopes represent central objects of study in the contemporary discrete convex geometry, on the crossroads of combinatorics, commutative algebra, and algebraic geometry. In Sections 2 and 3 we describe two very different possible ways of advancing the theory of normal polytopes to next essential level, invo...
متن کاملA convex combinatorial property of compact sets in the plane and its roots in lattice theory
K. Adaricheva and M. Bolat have recently proved that if $,mathcal U_0$ and $,mathcal U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $jin {0,1,2}$ and $kin{0,1}$ such that $,mathcal U_{1-k}$ is included in the convex hull of $,mathcal U_kcup({A_0,A_1, A_2}setminus{A_j})$. One could say disks instead of circles.Here we prove the existence of such a $j$ and $k$ ...
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...
متن کامل15 Basic 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 to linear and combinatorial optimization. In this chapter we try to give a short introduction, provide a sketch of ...
متن کامل