scholarly journals More on the Metric Projection onto a Closed Convex Set in a Hilbert Space

2016 ◽  
pp. 529-534
Author(s):  
Biagio Ricceri
Keyword(s):  
Hilbert Space ◽  
Convex Set ◽  
Metric Projection ◽  
Closed Convex Set

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.

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.

scholarly journals The projection operator in a Hilbert space and its directional derivative. Consequences for the theory of projected dynamical systems

2004 ◽  
Vol 2 (1) ◽  
pp. 71-95 ◽  
Author(s):  
George Isac ◽  
Monica G. Cojocaru
Keyword(s):  
Hilbert Space ◽  
Dynamical Systems ◽  
Representation Theorem ◽  
Projection Operator ◽  
Directional Derivative ◽  
Metric Projection ◽  
Pseudomonotone Operators ◽  
Projected Dynamical Systems ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space

In the first part of this paper we present a representation theorem for the directional derivative of the metric projection operator in an arbitrary Hilbert space. As a consequence of the representation theorem, we present in the second part the development of the theory of projected dynamical systems in infinite dimensional Hilbert space. We show that this development is possible if we use the viable solutions of differential inclusions. We use also pseudomonotone operators.

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.

Extreme Points in H1(R)

1967 ◽  
Vol 19 ◽  
pp. 312-320 ◽  
Author(s):  
Frank Forelli
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Convex Set ◽  
Complete Solution ◽  
Extreme Points ◽  
Open Unit ◽  
Open Unit Disk ◽  
Open Riemann Surface ◽  
Harmonic Majorant ◽  
Closed Convex Set

Let R be an open Riemann surface. ƒ belongs to H1(R) if ƒ is holomorphic on R and if the subharmonic function |ƒ| has a harmonie majorant on R. Let p be in R and define ||ƒ|| to be the value at p of the least harmonic majorant of |ƒ|. ||ƒ|| is a norm on the linear space H1(R), and with this norm H1(R) is a Banach space (7). The unit ball of H1(R) is the closed convex set of all ƒ in H1(R) with ||ƒ|| ⩽ 1. Problem: What are the extreme points of the unit ball of H1(R)? de Leeuw and Rudin have given a complete solution to this problem where R is the open unit disk (1).

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