Stability of best approximation of a convex body by a ball of fixed radius

The finite-dimensional problem of the best approximation (in the Hausdorff metric) of a convex body by a ball of arbitrary norm with a fixed radius is considered. The stability and sensitivity of the solution to errors in specifying the convex body to be approximated and the unit ball of the used norm are analyzed. It is shown that the solution of the problem is stable with respect to the functional and, if the solution is unique, the center of the best approximation ball is stable as well. The sensitivity of the solution to the error with respect to the functional is estimated (regardless of the radius of the ball). A sensitivity estimate for the center of the best approximation ball is obtained under the additional assumption that the estimated body and the ball of the used norm are strongly convex. This estimate is related to the range of radii of the approximating ball.     

Stability theorems for projections of convex sets

LetH ^n?1 denote the set of all (n ? 1)-dimensional linear subspaces of euclideann-dimensional spaceE ⁿ (n≧3), and letJ andK be two compact convex subsets ofE ⁿ. It is well-known thatJ andK are translation equivalent (or homothetic) if for allH ∈H ^n?1 the respective orthogonal projections, sayJ _H, K_H, are translation equivalent (or homothetic). This fact gives rise to the following stability problem: Ifε≧0 is given, and if for everyH ∈H ^n?1 a translate (or homothetic copy) ofK _H is within Hausdorff distanceε ofJ _H, how close isJ to a nearest translate (or homothetic copy) ofK? In the case of translations it is shown that under the above assumptions there is always a translate ofK within Hausdorff distance (1 + 2√2)ε ofJ. Similar results for homothetic transformations are proved and applications regarding the stability of characterizations of centrally symmetric sets and spheres are given. Furthermore, it is shown that these results hold even ifH ^n?1 is replaced by a rather small (but explicitly specified) subset ofH ^n?1.     

An example of a convex body without symmetric projections

Many crucial results of the asymptotic theory of symmetric convex bodies were extended to the non-symmetric case in recent years. That led to the conjecture that for everyn-dimensional convex bodyK there exists a projectionP of rankk, proportional ton, such thatPK is almost symmetric. We prove that the conjecture does not hold. More precisely, we construct ann-dimensional convex bodyK such that for everyk >C√nlnn and every projectionP of rankk, the bodyPK is very far from being symmetric. In particular, our example shows that one cannot expect a formal argument extending the “symmetric” theory to the general case. This author holds a Lady Davis Fellowship.     

Analytic functions whose range is dense in a ball

Denote by Δ(resp. $\text{Δ}$) the open (resp. closed) unit disc in C. Let E be a closed subset of the unit circle T and let F be a relatively closed subset of T ? E of Lesbesgue measure zero. The following result is proved. Given a complex Banach space X and a bounded continuous function f:F → X, there exists an extension f? of f, bounded and continuous on $\text{}\text{?}\text{gD ? E}$, analytic on Δ and satisfying sup$\text{{6}\text{f}\text{?}\text{(z)6:zε}\text{δ}\text{?E}$. This is applied to show that for any separable complex Banach space X there exists an analytic function from Δ to X whose range is contained and dense in the unit ball of X.     

Steered sequential projections for the inconsistent convex feasibility problem

We study a steered sequential gradient algorithm which minimizes the sum of convex functions by proceeding cyclically in the directions of the negative gradients of the functions and using steered step-sizes. This algorithm is applied to the convex feasibility problem by minimizing a proximity function which measures the sum of the Bregman distances to the members of the family of convex sets. The resulting algorithm is a new steered sequential Bregman projection method which generates sequences that converge if they are bounded, regardless of whether the convex feasibility problem is or is not consistent. For orthogonal projections and affine sets the boundedness condition is always fulfilled.     

The only convex body with extremal distance from the ball is the simplex

We can extend the Banach-Mazur distance to be a distance between non-symmetric sets by allowing affine transformations instead of linear transformations. It was proved in [J] that for every convex bodyK we haved(K, D)≤n. It is proved that ifK is a convex body in ℝ ⁿ such thatd(K, D)=n, thenK is a simplex. This article is an M.Sc. thesis written under the supervision of E. Gluskin and V.D. Milman at Tel Aviv University. Partially supported by a G.I.F. grant.     

A parallel subgradient projections method for the convex feasibility problem

An iterative method to solve the convex feasibility problem for a finite family of convex sets is presented. The algorithm consists in the application of a generalization of an acceleration procedure introduced by De Pierro, in a parallel version of the Subgradient Projections Method proposed by Censor and Lent. The generated sequence is shown to converge for any starting point. Some numerical results are presented.     

Inscribed ball and enclosing box methods for the convex maximization problem

Many important classes of decision models give rise to the problem of finding a global maximum of a convex function over a convex set. This problem is known also as concave minimization, concave programming or convex maximization. Such problems can have many local maxima, therefore finding the global maximum is a computationally difficult problem, since standard nonlinear programming procedures fail. In this article, we provide a very simple and practical approach to find the global solution of quadratic convex maximization problems over a polytope. A convex function achieves its global maximum at extreme points of the feasible domain. Since an inscribed ball does not contain any extreme points of the domain, we use the largest inscribed ball for an inner approximation while a minimal enclosing box is exploited for an outer approximation of the domain. The approach is based on the use of these approximations along with the standard local search algorithm and cutting plane techniques.     

Modified Shephard’s problem on projections of convex bodies

We disprove a conjecture of A. Koldobsky asking whether it is enough to compare (n − 2)-derivatives of the projection functions of two symmetric convex bodies in the Shephard problem in order to get a positive answer in all dimensions. The author was supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN-511953     

Method for finding an approximate solution of the asphericity problem for a convex body

Given a convex body, the finite-dimensional problem is considered of minimizing the ratio of its circumradius to its inradius (in an arbitrary norm) by choosing a common center of the circumscribed and inscribed balls. An approach is described for obtaining an approximate solution of the problem, whose accuracy depends on the error of a preliminary polyhedral approximation of the convex body and the unit ball of the used norm. The main result consists of developing and justifying a method for finding an approximate solution with every step involving the construction of supporting hyperplanes of the convex body and the unit ball of the used norm at some marginal points and the solution of a linear programming problem.     

Linearly ordered groups whose system of convex subgroups is central

The order P on a group G is called rigid if for p P the relation p¦[x, p]¦ P holds for every x G, =±1. In this note we give criteria for the existence of a rigid linear order, for the extendability of a rigid partial order to a rigid linear order, and for the extendability of each rigid partial order to a rigid linear order on a group. It is proved that the class of groups each of whose rigid partial orders can be extended to a rigid linear order is closed with respect to direct products. A new proof of the theorem of M. I. Kargapolov which states that if a group G can be approximated by finite p-groups for infinite number of primes p, then it has a central system of subgroups with torsion-free factors is presented.Translated from Matematicheskie Zametki, Vol. 19, No. 1, pp. 85–90, January, 1976.     

The dirichlet problem in a domain whose boundary is partly degenerated

Summary In this paper we bring a formulation of the Dirichlet problem for strongly elliptic equations in domains vhose boundaries may include manifolds of different dimensions. It is shown that, under certain regularity conditions, this problem is equivalent to the generalized Dirichlèt problem, with respect to existence and uniquennes of solutions. This paper represents part of a thesis submitted to the Senate of the Technion, Israel Institute of Technology, in partial fulfillment of the requirements for the degree of Doctor of Science. The author wishes to tank ProfessorS. Agmon for his gudance and help in the preparation of this work.     

Exceptional directions for a convex body

Let K be a convex body in Rⁿ andO be a point inside K. We examine the Grassmann manifold of k-planes passing throughO. We take as exceptional the planes intersecting K along a body having at least one (k – 1)-dimensional face such that it does not have points inside the hyperfaces of body K. We prove that in the Grassmann manifold G _k ⁿ the set of such exceptional planes is of measure zero.Translated from Matematicheskie Zametki, Vol. 20, No. 3, pp. 365–371, September, 1976.The author thanks V. A. Zalgaller for aid and advice on the work.     

Stability problems in certain inverse problems of reconstruction of convex compacta from their projections

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 33, No. 3, pp. 50–57, May–June, 1992.     

Radial solutions for a Dirichlet problem in a ball

The Ambrosetti-Prodi boundary value problem with an asymptotically linear nonlinearity is considered. Under general conditions on the nonlinearity it is shown that there exist positive and negative solutions. In the case when the domain is a ball in Rⁿ and the nonlinearity “crosses” the first n eigenvalues, corresponding to radial eigenfunctions, it is proved that there are at least n + 1 radial solution.