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.

Counterexamples Concerning Support Theorems for Convex Sets in Hilbert Space

1988 ◽  
Vol 31 (1) ◽  
pp. 121-128 ◽  
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.

Rates of convergence for random approximations of convex sets

1996 ◽  
Vol 28 (02) ◽  
pp. 384-393 ◽  
Author(s):  
Lutz Dümbgen ◽  
Günther Walther
Keyword(s):  
Convex Subset ◽  
Hausdorff Distance ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Almost Sure Convergence ◽  
Rates Of Convergence ◽  
Random Sets ◽  
Compact Convex Set

The Hausdorff distance between a compact convex set K ⊂ ℝd and random sets is studied. Basic inequalities are derived for the case of being a convex subset of K. If applied to special sequences of such random sets, these inequalities yield rates of almost sure convergence. With the help of duality considerations these results are extended to the case of being the intersection of a random family of halfspaces containing K.

scholarly journals Further inequalities for convex sets with lattice point constraints in the plane

1980 ◽  
Vol 21 (1) ◽  
pp. 7-12 ◽  
Author(s):  
P.R. Scott
Keyword(s):  
Lattice Point ◽  
Convex Sets ◽  
Convex Set ◽  
Integral Lattice ◽  
Image Position ◽  
Closed Convex Set

Let K be a bounded closed convex set in the plane containing no points of the integral lattice in its interior and having width w, area A, perimeter p and circumradius R. The following best possible inequalities are established:

SEPARATION OF CONVEX SETS IN EXTENDED NORMED SPACES

2015 ◽  
Vol 99 (2) ◽  
pp. 145-165 ◽  
Author(s):  
G. BEER ◽  
J. VANDERWERFF
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Separation Theorems ◽  
Normed Spaces ◽  
Finite Dimensional ◽  
Weakly Closed ◽  
Separation Of Convex Sets ◽  
Real Linear ◽  
Closed Convex Set ◽  
Special Case

We give continuous separation theorems for convex sets in a real linear space equipped with a norm that can assume the value infinity. In such a space, it may be impossible to continuously strongly separate a point $p$ from a closed convex set not containing $p$, that is, closed convex sets need not be weakly closed. As a special case, separation in finite-dimensional extended normed spaces is considered at the outset.

scholarly journals Farthest Points and Subdifferential in -Normed Spaces

2008 ◽  
Vol 2008 ◽  
pp. 1-6
Author(s):  
S. Hejazian ◽  
A. Niknam ◽  
S. Shadkam
Keyword(s):  
Compact Subset ◽  
Normed Space ◽  
Dense Subset ◽  
Normed Spaces ◽  
Point Mapping ◽  
Farthest Points ◽  
Farthest Point ◽  
In Virtue Of

We study the farthest point mapping in a -normed space in virtue of subdifferential of , where is a weakly sequentially compact subset of . We show that the set of all points in which have farthest point in contains a dense subset of .

Numerical Range and Convex Sets*

1974 ◽  
Vol 17 (2) ◽  
pp. 295-296 ◽  
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.

scholarly journals Two inequalities for convex sets in the plane

1978 ◽  
Vol 19 (1) ◽  
pp. 131-133 ◽  
Author(s):  
P.R. Scott
Keyword(s):  
Euclidean Plane ◽  
Convex Sets ◽  
Convex Set ◽  
Image Position ◽  
Closed Convex Set

Let K be a bounded, closed, convex set in the euclidean plane having diameter d, width w, inradius r, and circumradius R. We show thatandwhere both these inequalities are best possible.

scholarly journals Gateaux Differentiability of Convex Functions and Weak Dentable Set in Nonseparable Banach Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Shaoqiang Shang ◽  
Yunan Cui
Keyword(s):  
Convex Function ◽  
Banach Spaces ◽  
Convex Functions ◽  
Convex Set ◽  
Dense Subset ◽  
Lower Semicontinuous ◽  
Asplund Space ◽  
Bounded Subset ◽  
Exposed Point ◽  
Closed Convex Set

In this paper, we prove that if C⁎⁎ is a ε-separable bounded subset of X⁎⁎, then every convex function g≤σC is Ga^teaux differentiable at a dense Gδ subset G of X⁎ if and only if every subset of ∂σC(0)∩X is weakly dentable. Moreover, we also prove that if C is a closed convex set, then dσC(x⁎)=x if and only if x is a weakly exposed point of C exposed by x⁎. Finally, we prove that X is an Asplund space if and only if, for every bounded closed convex set C⁎ of X⁎, there exists a dense subset G of X⁎⁎ such that σC⁎ is Ga^teaux differentiable on G and dσC⁎(G)⊂C⁎. We also prove that X is an Asplund space if and only if, for every w⁎-lower semicontinuous convex function f, there exists a dense subset G of X⁎⁎ such that f is Ga^teaux differentiable on G and df(G)⊂X⁎.

scholarly journals A family of inequalities for convex sets

1979 ◽  
Vol 20 (2) ◽  
pp. 237-245 ◽  
Author(s):  
P.R. Scott
Keyword(s):  
Euclidean Plane ◽  
Convex Sets ◽  
Convex Set ◽  
Upper Bounds ◽  
Image Position ◽  
Closed Convex Set

Let K be a bounded, closed convex set in the euclidean plane. We denote the diameter, width, perimeter, area, inradius, and circumradius of K by d, w, p, A, r, and R respectively. We establish a number of best possible upper bounds for (w−2r)d, (w−2r)R,(w−2r)p, (w−2r)A in terms of w and r. Examples are:

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