General Convex Sets Research Articles

The Stop Operator Related to a Convex Polyhedron

Polyhedra, bounded and unbounded, play a special role as domains for the stop operator: Unlike in general convex sets, in a polyhedron the stop operator is always globally Lipschitz in W1, 1, and it maps periodic forcing functions to solutions converging to a periodic solution.

Regularity of solutions for arbitrary order variational inequalities with general convex sets

We consider the regularity of solutions of a system for nonlinear arbitrary order variational inequalities with some general concrete closed convex sets. The maximal function method and the convergence method are used, and a higher integrability of the derivatives is obtained.

Minimizing energy on locally compact spaces: existence and approximation

In this paper we study the existence and approximation of solutions of the energy minimization problem on general convex sets of positive Radon measures by using direct methods. To establish the existence of solutions we mainly use the vague topology and weak hypotheses on the kernel, for example one of them is the existence of vague cluster points to Cauchy sequences. Furthermore, we give sufficient conditions on the convex set to assure the existence of minimizing sequences of discrete measures. This can be thought of as a generalization of the Transfinite Diameter method and of the Frotsman techniques. In addition, when the kernel is extended continuously the computation of the minimal energy or of minimizing sequences is operative. Finally, we obtain regular capacities on the class of the positive Radon measures, for which every subset is capacitable

Shelling pseudopolyhedra

The notion of shellability originated in the context of polyhedral complexes and combinatorial topology. An abstraction of this concept for graded posets (i.e., graded partially ordered sets) was recently introduced by Björner and Wachs first in the finite case [1] and then with Walker in the infinite case [11]. Many posets arising in combinatorics and in convex geometry were investigated and some proved to be shellable. A key achievement was the proof by Bruggesser and Mani that boundary complexes of convex polytopes are shellable [4]. We extend here the result of Bruggesser and Mani to polyhedral complexes arising as boundary complexes of more general convex sets, called pseudopolyhedra, with suitable asymptotic behavior. This includes a previous result on tilings of a Euclidean space ℝ d which are projections of the boundary of a (d+1)-pseudopolyhedron [7].

Computing the distance between general convex objects in three-dimensional space

A methodology for computing the distance between objects in three-dimensional space is presented. The convex polytope is replaced by a general convex set, avoiding the errors caused by the usual polytope approximations and actually reducing the overall computational time. The basic algorithm is a simple extension of the polytope distance algorithm described by E.G. Gilbert et al. (1988). It utilizes the support mappings of the sets representing the objects. A calculus for evaluating these mappings that allows the extended algorithm to be applied to a rich family of nonpolytopal objects is presented. While the convergence of the algorithm is not finite, it is fast and an effective stopping condition that guarantees the accuracy of the numerical solution is available. Extensive numerical experiments support the claimed efficiency.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Removable sets of support points of convex sets in Banach spaces

A corollary of the Bishop-Phelps theorem is that a closed convex subset C of a Banach space can always be represented as the intersection of its supporting closed half-spaces. In this paper an investigation is made of those subsets S of C such that C is the intersection of those closed half-spaces which support it at points of C\S. This will be true for sets S which are small relative to C, where smallness can be measured in terms of dimension, density character, or a-compactness. Suppose that C is a nonempty closed convex subset of a Banach space E. A point x E C is called a support point of C if there exists a nonzero functional f E E* which attains its supremum on C at x. Any such functional is said to be a support functional of C and the set of all support points is denoted by supp C. It is known [1] that the support points of C are always dense in bdry C, the boundary of C, and that the support functionals of C are norm dense among those which are bounded above on C. A corollary of the methods used for these results is the fact that C is always the intersection of all those closed half-spaces which are defined by support functionals [1, Corollary 2]. This result is trivial, of course, if C has nonempty interior, since every boundary point of C is a support point. In this case, in fact, it is easily seen that C can be represented as the intersection of those halfspaces which support it at the points of D for any dense subset D of bdry C (see part (iv) of Theorem 1, below). This fact has played a key role in characterizing those generators of Co-semigroups of operators which leave invariant a given closed convex set with interior [2, 3]. In considering the extension of his work [2] to more general convex sets, K. N. Boyadzhiev raised the question (in a letter to the author) of whether one could express C as the intersection of those closed half-spaces which support C at some proper subset of supp C. The purpose of this note is to give some answers to this question. A little thought shows that one has to use some care in deleting subsets of supp C. For instance, if C is a line segment, then a point x of E\C on the line determined by C can be separated from C only by those support functionals which attain their maximum on C at the endpoint nearest x; that is, one cannot remove that endpoint from supp C and still separate x from C by a support functional. (If the dimension of E is at least two, then every point of C is a support point, so supp C minus a single poinit is still dense in bdry C.) If C is infinite dimensional, then one can remove a finite subset S of supp C; in fact, as we show below, one can remove a finite dimensional subset and still obtain C as the intersection of half-spaces which Received by the editors October 7, 1985 and, in revised form, January 20, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 46B20, 47D05.

Some Fundamental Properties of Kolmogorov-Smirnov Consonance Sets

Consonance sets are obtained by inverting inequalities for goodness-of-fit statistics which are considered as functions of the parameters of a model distribution. In this paper we present some evidence to indicate that the use of the Kolmogorov-Smimov statistic assures reasonable results for location and scale parameters. Specifically, the consonance sets obtained are convex and finite. The development also provides a method for set construction. An example shows that the discrete X2 statistic does not in general yield convex sets.

Reducing Concave Programs with Some Linear Constraints

The problem of maximizing a concave function over a general convex set subject to linear inequality constraints is reduced to a finite sequence of sub-problems involving linear equality constraints. This reduction can be expected to be computationally useful when there are but a few constraints, or when at most a few constraints are binding at the optimal solution of the original problem, or when prior (though possibly fallible) information is available concerning which constraints are likely to be binding. For quadratic programs the procedure specializes to an improved version of the Theil-van de Panne method. Computational considerations and experience are discussed, and a graphical example is given. The theory and viewpoint developed herein provide the foundation for related reduction procedures that may prove computationally useful even for large problems in the absence of a priori information.

Generalized convex sets in the plane

Generalized convex sets in the plane