The distance between two convex sets in Hilbert space

We apply the generalised concept of witness operators to arbitrary convex sets, and review the criteria for the optimisation of these general witnesses. We then define an embedding of state vectors and operators into a higher-dimensional Hilbert space. This embedding leads to a connection between any Schmidt number witness in the original Hilbert space and a witness for Schmidt number two (i.e. the most general entanglement witness) in the appropriate enlarged Hilbert space. Using this relation we arrive at a conceptually simple method for the construction of Schmidt number witnesses in bipartite systems.

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Counterexamples Concerning Support Theorems for Convex Sets in Hilbert Space

Canadian Mathematical Bulletin ◽

10.4153/cmb-1988-019-1 ◽

1988 ◽

Vol 31 (1) ◽

pp. 121-128 ◽

Cited By ~ 6

Author(s):

R. R. Phelps

Keyword(s):

Banach Space ◽

Hilbert Space ◽

Convex Function ◽

Convex Subset ◽

Convex Sets ◽

Lower Semicontinuous ◽

Nonempty Closed Convex Subset ◽

Common Point ◽

Support Points ◽

Support Theorems

AbstractThe Bishop-Phelps theorem guarantees the existence of support points and support functionals for a nonempty closed convex subset of a Banach space; equivalently, it guarantees the existence of subdifferentials and points of subdifferentiability of a proper lower semicontinuous convex function on a Banach space. In this note we show that most of these results cannot be extended to pairs of convex sets or functions, even in Hilbert space. For instance, two proper lower semicontinuous convex functions need not have a common point of subdifferentiability nor need they have a subdifferential in common. Negative answers are also obtained to certain questions concerning density of support points for the closed sum of two convex subsets of Hilbert space.

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Porosity and the bounded linear regularity property

Journal of Applied Analysis ◽

10.1515/jaa-2014-0001 ◽

2014 ◽

Vol 20 (1) ◽

pp. 1-6 ◽

Cited By ~ 5

Author(s):

Simeon Reich ◽

Alexander J. Zaslavski

Keyword(s):

Hilbert Space ◽

Convex Sets ◽

Regularity Property ◽

Nonempty Intersection ◽

Normed Spaces ◽

Infinite Products ◽

Bounded Linear ◽

Linear Regularity ◽

Nearest Point ◽

Convex Subsets

Abstract.H. H. Bauschke and J. M. Borwein showed that in the space of all tuples of bounded, closed, and convex subsets of a Hilbert space with a nonempty intersection, a typical tuple has the bounded linear regularity property. This property is important because it leads to the convergence of infinite products of the corresponding nearest point projections to a point in the intersection. In the present paper we show that the subset of all tuples possessing the bounded linear regularity property has a porous complement. Moreover, our result is established in all normed spaces and for tuples of closed and convex sets, which are not necessarily bounded.

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The method of cyclic projections for closed convex sets in Hilbert space

Recent Developments in Optimization Theory and Nonlinear Analysis - Contemporary Mathematics ◽

10.1090/conm/204/02620 ◽

1997 ◽

pp. 1-38 ◽

Cited By ~ 79

Author(s):

Heinz H. Bauschke ◽

Jonathan M. Borwein ◽

Adrian S. Lewis

Keyword(s):

Hilbert Space ◽

Convex Sets

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Numerical Range and Convex Sets*

Canadian Mathematical Bulletin ◽

10.4153/cmb-1974-058-3 ◽

1974 ◽

Vol 17 (2) ◽

pp. 295-296 ◽

Cited By ~ 3

Author(s):

Fredric M. Pollack

Keyword(s):

Hilbert Space ◽

Linear Operator ◽

Convex Subset ◽

Bounded Linear Operator ◽

Convex Sets ◽

Numerical Range ◽

Bounded Subset ◽

Bounded Linear ◽

Empty Convex ◽

Image Position

The numerical range W(T) of a bounded linear operator T on a Hilbert space H is defined byW(T) is always a convex subset of the plane [1] and clearly W(T) is bounded since it is contained in the ball of radius ‖T‖ about the origin. Which non-empty convex bounded subsets of the plane are the numerical range of an operator? The theorem we prove below shows that every non-empty convex bounded subset of the plane is W(T) for some T.

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