An algorithm for determining all extreme points of a convex polytope

An algorithm for determining all the extreme points of a convex polytope associated with a set of linear constraints, via the computation of basic feasible solutions to the constraints, is presented. The algorithm is based on the product-form revised simplex method and as such can be readily linked onto standard linear programming codes. Applications of such an algorithm are reviewed and limited computational experience given.     

Method of orienting curves for determining the convex hull of a finite set of points in the plane

In this article, we present an efficient algorithm to determine the convex hull of a finite planar set using the idea of the Method of Orienting Curves (introduced by Phu in Zur Lösung einer regulären Aufgabenklasse der optimalen Steuerung in Großen mittels Orientierungskurven, Optimization, 18 (1987), pp. 65–81, for solving optimal control problems with state constraints). The convex hull is determined by parts of orienting lines and a final line. Two advantages of this algorithm over some variations of Graham's convex hull algorithm are presented.     

Applications of a semi-dynamic convex hull algorithm

We obtain new results for manipulating and searching semi-dynamic planar convex hulls (subject to deletions only), and apply them to derive improved bounds for two problems in geometry and scheduling. The new convex hull results are logarithmic time bounds for set splitting and for finding a tangent when the two convex hulls are not linearly separated. Using these results, we solve the following two problems optimally inO(n logn) time: (1) [matching] givenn red points andn blue points in the plane, find a matching of red and blue points (by line segments) in which no two edges cross, and (2) [scheduling] givenn jobs with due dates, linear penalties for late completion, and a single machine on which to process them, find a schedule of jobs that minimizes the maximum penalty.     

The convex hull of the lattice points inside a curve

Let $C$ be a smooth convex closed plane curve. The $C$ -ovals $C(R,u,v)$ are formed by expanding by a factor  $R$ , then translating by  $(u,v)$ . The number of vertices $V(R,u,v)$ of the convex hull of the integer points within or on  $C(R,u,v)$ has order  $R^{2/3}$ (Balog and Bárány) and has average size $BR^{2/3}$ as $R$ varies (Balog and Deshouillers). We find the space average of  $V(R,u,v)$ over vectors  $(u,v)$ to be  $BR^{2/3}$ with an explicit coefficient  $B$ , and show that the average over  $R$ has the same  $B$ . The proof involves counting edges and finding the domain $D(q,a)$ of an integer vector, the set of  $(u,v)$ for which the convex hull has a directed edge parallel to  $(q,a)$ . The resulting sum over bases of the integer lattice is approximated by a triple integral.     

On triangulations of the convex hull ofn points

A setS ofn points in Euclideand-space determines a convex hull which can be triangulated into some numberm of simplices using the points ofS as vertices. We characterize those setsS for which all triangulations minimizem. This is used to characterize sets of points maximizing the volume of the smallest non-trivial simplex. This work was supported in part by NSF Grants MCS 81-02519 and MCS 82-03347. This work supported in part by NSF Grants MCS 81-02519 and MCS 82-03347 Dedicated to Paul Erdős on his seventieth birthday     

Quadratic problems defined on a convex hull of points

In this paper we consider an algorithm for a class of quadratic problems defined on a polytope which is described as the convex hull of a set of points. The algorithm is based on simplex partitions using convex underestimating functions.     

On the convex hull of uniform random points in a simpled-polytope

Denote the expected number of facets and vertices and the expected volume of the convex hullP _n ofn random points, selected independently and uniformly from the interior of a simpled-polytope byE _n (f), E _n (v), andE _n (V), respectively. In this note we determine the sharp constants of the asymptotic expansion ofE _n (f), E _n (v), andE _n (V), asn tends to infinity. Further, some results concerning the expected shape ofP _n are given. The work of F. Affentranger was supported by the Swiss National Foundation.     

Distribution-independent properties of the convex hull of random points

Denote byV _n ^(d) the expected volume of the convex hull ofn points chosen independently according to a given probability measure in Euclideand-spaceE ^d. Ifd=2 ord=3 and is the measure corresponding to the uniform distribution on a convex body inE ^d, Affentranger and Badertscher derived that

On the shape of the convex hull of random points

Summary Denote by E _n the convex hull of n points chosen uniformly and independently from the d-dimensional ball. Let Prob(d, n) denote the probability that E _n has exactly n vertices. It is proved here that Prob(d, 2 ^d/2 d ^-)1 and Prob(d, 2 ^d/2 d ^(3/4)+)0 for every fixed >0 when d. The question whether E _n is a k-neighbourly polytope is also investigated.     

Extreme points of the closed convex hull of composition operators

We study the extreme points of the closed convex hull of the set of all composition operators on the space of bounded analytic functions and the disk algebra.     

A simple algorithm for building the 3-D convex hull

A new algorithm is given for finding the convex hull of a finite set of distinct points in three-dimensional space. The algorithm finds the faces of the hull one by one, thus gradually building the polyhedron that constitutes the hull. The algorithm is described as developed through stepwise refinement.     

On convex bodies containing a given number of lattice points

Given a convex body K in Euclidean space, a necessary and sufficient condition is established in order that for each n there exists a homothetic copy of K containing exactly n lattice points. Similar theorems are proved for congruent copies of K and for discrete sets other than lattices.     

The order of points on the second convex hull of a simple polygon

Given a setV of points on the plane, if {q ₁,...,q _n } is the set of points on the second convex hull ofV, the order in which these points are visited in anyV-gon is characterized. This order must verify two similar conditions to those of Kuratowski's theorem for planar graphs. Moreover, the number of possible orders that verify these conditions is obtained. It isO(5 ⁿ ).     

Equiaffine inner parallel curves of a plane convex body and the convex hulls of randomly chosen points

Summary. A general formula is proved, which relates the equiaffine inner parallel curves of a plane convex body and the probability that the convex hull of j independent random points is disjoint from the convex hull of k further independent random points. This formula is applied to improve some well-known results in geometric probability. For example, an estimate, which was established for a special case by L. C. G. Rogers, is obtained with the best possible bound, and an asymptotic formula due to A. Rényi and R.␣Sulanke is extended to an asymptotic expansion. Received: 21 May 1996     

On compact convex sets with given extreme points

The research was financially supported by the Russian Foundation for Fundamental Research (Grant 94-01-01286).     

An interior points algorithm for the convex feasibility problem

The problem of finding a point in the intersection of a finite family of convex sets in the Euclidean space R″ is considered here. We present a general algorithmic scheme which employs projections onto separating hyperplanes instead of projections onto the convex sets. This scheme includes the method of successive projections of Gubin et al., USSR Comp. Math. and Math. Phys. 7 (1967), 1–24, as a special case. A different realization proposed here is capable of handling the problem when the sets are solid and an interior point of each set is available. This alternative algorithm may, in certain cases, be more attractive than the method of Gubin et al.     

Interior points of the convex hull of few points in\mathbb{E}^d

We show that ifP , |P|=d+k,dk1 andO int convP, then there exists a simplexS of dimension with vertices inP, satisfyingO rel intS, the bound being sharp. We give an upper bound for the minimal number of vertices of facets of a (j-1)-neighbourly convex polytope in withv vertices.Research (partially) supported by Hung. Nat. Found. for Sci. Research, grant no. 1817Research (partially) supported by Hung. Nat. Found. for Sci. Research, grant no. 326-0213     

Delaunay triangulation and the convex hull ofn points in expected linear time

An algorithm is presented which produces a Delaunay triangulation ofn points in the Euclidean plane in expected linear time. The expected execution time is achieved when the data are (not too far from) uniformly distributed. A modification of the algorithm discussed in the appendix treats most of the non-uniform distributions. The basis of this algorithm is a geographical partitioning of the plane into boxes by the well-known Radix-sort algorithm. This partitioning is also used as a basis for a linear time algorithm for finding the convex hull ofn points in the Euclidean plane.