Best Approximation from the Intersection of a Closed Convex Set and a Polyhedron in Hilbert Space, Weak Slater Conditions, and the Strong Conical Hull Intersection Property

1999 ◽  
Vol 10 (1) ◽  
pp. 252-268 ◽  
Author(s):  
Frank Deutsch ◽  
Wu Li ◽  
Joseph D. Ward
Keyword(s):  
Hilbert Space ◽  
Best Approximation ◽  
Convex Set ◽  
Intersection Property ◽  
Closed Convex Set ◽  
Conical Hull

scholarly journals Characterisation of normed linear spaces with Mazur's intersection property

1978 ◽  
Vol 18 (1) ◽  
pp. 105-123 ◽  
Author(s):  
J.R. Giles ◽  
D.A. Gregory ◽  
Brailey Sims
Keyword(s):  
Euclidean Space ◽  
Convex Set ◽  
Intersection Property ◽  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Strongly Exposed Points ◽  
Denting Points ◽  
Mazur's Intersection Property ◽  
Closed Convex Set ◽  
Space Property

Normed linear spaces possessing the euclidean space property that every bounded closed convex set is an intersection of closed balls, are characterised as those with dual ball having weak* denting points norm dense in the unit sphere. A characterisation of Banach spaces whose duals have a corresponding intersection property is established. The question of the density of the strongly exposed points of the ball is examined for spaces with such properties.

Uniform Mazur's Intersection Property of Balls

1987 ◽  
Vol 30 (4) ◽  
pp. 455-460 ◽  
Author(s):  
J. H. M. Whitfield ◽  
V. Zizler
Keyword(s):  
Banach Space ◽  
Convex Set ◽  
Closed Ball ◽  
Intersection Property ◽  
Mazur's Intersection Property ◽  
Closed Convex Set

AbstractWe give a dual characterization of the following uniformization of the Mazur's intersection property of balls in a Banach space X: for every ∊ > 0 there is a K > 0 such that whenever a closed convex set C ⊂ X and a point p ∊ X are such that diam C ≤ 1/∊ and dist(p, C) ≤ ∊, then there is a closed ball B of radius ≤ K with B ⊃ C and dist(p,B) ≥ ∊/2.

scholarly journals Properties of typical bounded closed convex sets in Hilbert space

2005 ◽  
Vol 2005 (4) ◽  
pp. 423-436
Author(s):  
F. S. de Blasi ◽  
N. V. Zhivkov
Keyword(s):  
Hilbert Space ◽  
Positive Integer ◽  
Convex Subset ◽  
Convex Sets ◽  
Convex Set ◽  
Dense Subset ◽  
Baire Category ◽  
Point Mapping ◽  
Farthest Point ◽  
Closed Convex Set

For a nonempty separable convex subsetXof a Hilbert spaceℍ(Ω), it is typical (in the sense of Baire category) that a bounded closed convex setC⊂ℍ(Ω)defines anm-valued metric antiprojection (farthest point mapping) at the points of a dense subset ofX, whenevermis a positive integer such thatm≤dimX+1.

Constrained best approximation in Hilbert space

1990 ◽  
Vol 6 (1) ◽  
pp. 35-64 ◽  
Author(s):  
Charles K. Chui ◽  
Frank Deutsch ◽  
Joseph D. Ward
Keyword(s):  
Hilbert Space ◽  
Best Approximation ◽  
Constrained Best Approximation

scholarly journals Auerbach Bases and Minimal Volume Sufficient Enlargements

2011 ◽  
Vol 54 (4) ◽  
pp. 726-738
Author(s):  
M. I. Ostrovskii
Keyword(s):  
Linear Space ◽  
Isometric Embedding ◽  
Normed Linear Space ◽  
Convex Set ◽  
Linear Projection ◽  
Minimal Volume ◽  
Finite Dimensional ◽  
Dimensional Normed Space ◽  
Closed Convex Set

AbstractLet BY denote the unit ball of a normed linear space Y. A symmetric, bounded, closed, convex set A in a finite dimensional normed linear space X is called a sufficient enlargement for X if, for an arbitrary isometric embedding of X into a Banach space Y, there exists a linear projection P: Y → X such that P(BY ) ⊂ A. Each finite dimensional normed space has a minimal-volume sufficient enlargement that is a parallelepiped; some spaces have “exotic” minimal-volume sufficient enlargements. The main result of the paper is a characterization of spaces having “exotic” minimal-volume sufficient enlargements in terms of Auerbach bases.

Points of egress in problems of Hamiltonian dynamics

1991 ◽  
Vol 109 (2) ◽  
pp. 405-417 ◽  
Author(s):  
C. J. Amick ◽  
J. F. Toland
Keyword(s):  
Convex Set ◽  
Hamiltonian Dynamics ◽  
Empty Interior ◽  
Natural Assumption ◽  
Initial Value ◽  
Lipschitz Continuous ◽  
Convexity Assumptions ◽  
Image Position ◽  
Closed Convex Set ◽  
Well Posed

First we consider an elementary though delicate question about the trajectory in ℝn of a particle in a conservative field of force whose dynamics are governed by the equationHere the potential function V is supposed to have Lipschitz continuous first derivative at every point of ℝn. This is a natural assumption which ensures that the initial-value problem is well-posed. We suppose also that there is a closed convex set C with non-empty interior C° such that V ≥ 0 in C and V = 0 on the boundary ∂C of C. It is noteworthy that we make no assumptions about the degeneracy (or otherwise) of V on ∂C (i.e. whether ∇V = 0 on ∂C, or not); thus ∂C is a Lipschitz boundary because of its convexity but not necessarily any smoother in general. We remark also that there are no convexity assumptions about V and nothing is known about the behaviour of V outside C.

On two pursuit differential game problems with state and geometric constraints in a Hilbert space

2021 ◽  
Vol 65 (3) ◽  
pp. 5-16
Author(s):  
Abbas Ja’afaru Badakaya ◽  
Keyword(s):  
Differential Game ◽  
Convex Set ◽  
Sufficient Conditions ◽  
Nonempty Closed Convex Subset ◽  
Geometric Constraints ◽  
First Order ◽  
Control Functions ◽  
Closed Convex Set ◽  
The Given ◽  
First Order Differential Equations

This paper concerns with the study of two pursuit differential game problems of many pursuers and many evaders on a nonempty closed convex subset of R^n. Throughout the period of the games, players must stay within the given closed convex set. Players’ laws of motion are defined by certain first order differential equations. Control functions of the pursuers and evaders are subject to geometric constraints. Pursuit is said to be completed if the geometric position of each of the evader coincides with that of a pursuer. We proved two theorems each of which is solution to a problem. Sufficient conditions for the completion of pursuit are provided in each of the theorems. Moreover, we constructed strategies of the pursuers that ensure completion of pursuit.

On Convex Fundamental Regions for a Lattice

1961 ◽  
Vol 13 ◽  
pp. 177-178 ◽  
Author(s):  
A. M. Macbeath
Keyword(s):  
Lebesgue Measure ◽  
Convex Set ◽  
Fundamental Region ◽  
Linearly Independent ◽  
Whole Space ◽  
Closed Convex Set ◽  
Set Of Points

Let ∧ be a lattice in Euclidean n-space, that is, ∧ is a set of points ε1a1+ … + εnanwherea1… ,anare linearly independent vectors and the ε run over all integers. Letμdenote the Lebesgue measure. A closed convex setFis called afundamental regionfor ∧ if the setsF+x (x∈∧) cover the whole space without overlapping; that is, ifF0is the interior ofF,and 0 ≠x∈ ∧, thenF0∩(F0+x) =ϕ.

Sign in / Sign up

Export Citation Format

Share Document