Approximative properties of sets in Hilbert space

Balaganskii, V. S.

doi:10.1007/bf01145720

Cited by 7 publications

(9 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Corollary 1 ( = Theorem 2 of [2]). If K is a nonconvex subset of H such that H\K is connected, then H\CK is uncountable unless K is the complement of an open ball Proof.…”

Section: £>0mentioning

confidence: 98%

“…also [18; p. 9]. Balaganskii [2], too, worked with the set T'K. Moreover, one can prove the following alternative representations of the set CK : CK = {x G H ; PK(x) is a singleton and PK is use at x} = {x G H ; <pK is Fréchet differentiable at x}.…”

Section: £>0mentioning

confidence: 99%

“…In these investigations, continuity was required at all points of H. In a paper of 1982, Balaganskii [2] considered the aspect of cardinality of the set of points of discontinuity of the metric projection. Typical for his results is the following theorem: If K is a Chebyshev set and if the set of points ofdiscontinuity of PK is countable, then K is convex.…”

Section: Introductionmentioning

confidence: 99%

“…Typical for his results is the following theorem: If K is a Chebyshev set and if the set of points ofdiscontinuity of PK is countable, then K is convex. The method of proof in [2] is an inversion method developed by Ficken and later used by Klee [12] and Asplund [1].…”

mentioning

confidence: 99%

See 3 more Smart Citations

On a property of metric projections onto closed subsets of Hilbert spaces

Frerking¹,

Westphal²

1989

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract.Applying the theory of monotone operators to the metric projection Pfc of a Hilbert space H onto a nonempty closed subset K of H we prove a kind of connectedness property of the set {x e H; Pk{x) is not a singleton or Pic is not upper semi-continuous at x} which is a typical set for investigations in best approximation. A result of Balaganskii is extended.Let H be a real Hilbert space with inner product (•, •) and norm || • || and let K be a nonempty closed subset of H. The metric projection PK : H -► 2K is defined by PK(x):={keK;\\k-x\\=dK(x)}, where dK: H -» R is the distance function of K. If PK(x) is a singleton for each x G H, then K is called a Chebyshev set.To motivate the purpose of this note let us first restrict ourselves to Chebyshev sets though our general result is concerned with arbitrary closed subsets of H.The problem of convexity of a Chebyshev set was studied by many authors under various conditions on the continuity of the metric projection, among others by Klee [11, 12], Vlasov [17] and Asplund [1]. In these investigations, continuity was required at all points of H. In a paper of 1982, Balaganskii [2] considered the aspect of cardinality of the set of points of discontinuity of the metric projection. Typical for his results is the following theorem: If K is a Chebyshev set and if the set of points ofdiscontinuity of PK is countable, then K is convex. The method of proof in [2] is an inversion method developed by Ficken and later used by Klee [12] and Asplund [1].In the present note we shall prove a more general result concerning a "connectedness" property of the set of discontinuities of the metric projection fromReceived by the editors May 31, 1988 and, in revised form, June 21, 1988. 1980 Mathematics Subject Classification (1985. Primary 41A65, 41A50; Secondary 47H05.Key words and phrases. Best approximation in Hilbert spaces, convexity of Chebyshev sets, continuity of metric projections, uniqueness in best approximation, monotony of metric projections.

show abstract

“…Corollary 1 ( = Theorem 2 of [2]). If K is a nonconvex subset of H such that H\K is connected, then H\CK is uncountable unless K is the complement of an open ball Proof.…”

Section: £>0mentioning

confidence: 98%

Section: £>0mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

On a property of metric projections onto closed subsets of Hilbert spaces

Frerking¹,

Westphal²

1989

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…For further contributions see [5] and [19]. Properties of the ambiguous loci of the metric projection have been studied by BALAGANSKII [1], BARTKE and BERENS [2], WESTPHAL and FRERKING [14] and VESEL~" [13].…”

Section: Notation and Introductionmentioning

confidence: 97%

Ambiguous loci of the metric projection onto compact starshaped sets in a Banach space

Blasi¹,

Kenderov

Myjak

1995

Monatshefte für Mathematik

View full text Add to dashboard Cite

Abstract. Let E be a real Banach space and 5P(E) the family of all nonempty compact starshaped subsets of E. Under the Hausdorff distance, So(E) is a complete metric space. The elements of the complement of a first Baire category subset of 5g(~:) are called typical elements of 5~(~). For XeS~(E) we denote by gx the metrical projection onto X, i.e. the mapping which associates to each a~: the set of all points in X closest to a. In this note we prove that, if E is strictly convex and separable with dim ~/> 2, then for a typical X E 5~(~) the map nx is not single valued at a dense set of points. Moreover, we show that a typical element of 5~(E) has kernel consisting of one point and set of directions dense in the unit sphere of E.

show abstract

Continuities of Metric Projection and GeometricConsequences

Huotari

1997

Journal of Approximation Theory

View full text Add to dashboard Cite

We discuss the geometric characterization of a subset K of a normed linear space via continuity conditions on the metric projection onto K. The geometric properties considered include convexity, tubularity, and polyhedral structure. The continuity conditions utilized include semicontinuity, generalized strong uniqueness and the non-triviality of the derived mapping. In finite-dimensional space with the uniform norm we show that convexity is equivalent to rotation-invariant almost convexity and we characterize those sets every rotation of which has continuous metric projection. We show that polyhedral structure underlies generalized strong uniqueness of the metric projection.1997 Academic Press, Inc.

show abstract

Approximative properties of sets in Hilbert space

Cited by 7 publications

References 5 publications

On a property of metric projections onto closed subsets of Hilbert spaces

On a property of metric projections onto closed subsets of Hilbert spaces

Ambiguous loci of the metric projection onto compact starshaped sets in a Banach space

Continuities of Metric Projection and GeometricConsequences

Contact Info

Product

Resources

About