“Almost exact” estimates for the Delone triangulation numbers are given. In particular lim n→∞ deln−1 n = 1 2πe = 0.0585498.... 1991 Mathematics Subject Classification. 52A22. The first named author was partially supported by the KBN grant no. 2 P03A 022 15. Both authors were partially supported by the Erwin-Schrödinger-Institute in Vienna. Typeset by AMS-TEX 1 2 PIOTR MANKIEWICZ CARSTEN SCHÜTT...

We investigate geometric and topological properties of $d$-majorization -- a generalization classical majorization to positive weight vectors $d \in \mathbb{R}^n$. In particular, we derive new, simplified characterization which allows us work out halfspace description the corresponding polytopes. That is, write set all are $d$-majorized by some given vector $y \mathbb{R}^n$ as an intersection f...

Given a combinatorial optimization problem and a subset N of natural numbers, we obtain a cardinality constrained version of this problem by permitting only those feasible solutions whose cardinalities are elements of N . In this paper we briefly touch on questions that addresses common grounds and differences of the complexity of a combinatorial optimization problem and its cardinality constra...

Given an n-dimensional convex body by a membership oracle in general, it is known that any polynomial-time deterministic algorithm cannot approximate its volume within ratio (n/ logn). There is a substantial progress on randomized approximation such as Markov chain Monte Carlo for a highdimensional volume, and for many #P-hard problems, while some deterministic approximation algorithms are rece...

We show that the minimum number of distinct edge-directions of a convex polytope with n vertices in Rd is θ(dn1/(d−1)).

Let K be a convex body in R. A random polytope is the convex hull [x1, ..., xn] of finitely many points chosen at random in K. E(K,n) is the expectation of the volume of a random polytope of n randomly chosen points. I. Bárány showed that we have for convex bodies with C3 boundary and everywhere positive curvature c(d) lim n→∞ vold(K)− E(K,n) ( vold(K) n ) 2 d+1 = ∫ ∂K κ(x) 1 d+1 dμ(x) where κ(...

In this paper we show that the diameter of a d-dimensional lattice polytope in [0, k]n is at most

We consider the Shaped Partition Problem of partitioning n given vectors in real k-space into p parts so as to maximize an arbitrary objective function which is convex on the sum of vectors in each part, subject to arbitrary constraints on the number of elements in each part. In addressing this problem, we study the Shaped Partition Polytope defined as the convex hull of solutions. The Shaped P...

Given a set N of items and a capacity b 2 IN, and let N j be the set of items with weight j, 1 j b. The 0/1 knapsack polytope is the convex hull of all 0/1 vectors that satisfy the inequality b X j=1 X i2N j jx i b: In this paper we rst present a complete linear description of the 0/1 knapsack polytope for two special cases: (a) N j = ; for all 1 < j b b 2 c and (b) N j = ; for all 1 < j b b 3 ...

We consider polyhedra with applications to well-know combinatorial optimization problems: the metric polytope mn and its relatives. For n ≤ 6 the description of the metric polytope is easy as mn has at most 544 vertices partitioned into 3 orbits; m7 the largest previously know instance has 275 840 vertices but only 13 orbits. Using its large symmetry group, we enumerate orbitwise 1 550 825 600 ...