Convex sets and nearest points

Phelps, R. R.

doi:10.1090/s0002-9939-1957-0087897-7

Cited by 90 publications

(37 citation statements)

References 8 publications

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“…It is possible to establish uniqueness as a consequence of strict convexity of the set that generates the wrapping. (For normed linear spaces see Lemma 3.2. of [19]) Proposition 3.7. Let (X, τ ) be a Hausdorff topological vector space with a convex neighborhood of the origin, C, whose closure does not contain halflines, and let A be a nonempty, convex and ξ C -proximinal subset of X.…”

Section: −Tmentioning

confidence: 95%

A topological approach to Best Approximation Theory

et al. 2004

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Abstract. The main goal of this paper is to put some light in several arguments that have been used through the time in many contexts of Best Approximation Theory to produce proximinality results. In all these works, the main idea was to prove that the sets we are considering have certain properties which are very near to the compactness in the usual sense. In the paper we introduce a concept (the wrapping) that allow us to unify all these results in a whole theory, where certain ideas from Topology are essential. Moreover, we do not only cover many of the known classical results but also prove some new results. Hence we prove that exists a strong interaction between General Topology and Best Approximation Theory. AMS Classification:41A65, 54-99

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Section: −Tmentioning

confidence: 95%

A topological approach to Best Approximation Theory

et al. 2004

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“…Definition 2.9 We say that X has Property (P) [9] if the nearest point map f shrinks distances whenever it exists for a closed convex set S ⊆ X.…”

Section: Definition 28mentioning

confidence: 99%

“…Metric projections have been very helpful in giving some partial answers of this problem. Phelps [9] showed that in an inner product space (in a strictly convex normed linear space), a Chebyshev set is convex if the associated metric projection is non-expansive. Here we extend this result to metric spaces.…”

Section: Introductionmentioning

confidence: 99%

A note on the convexity of Chebyshev sets

Narang

Sangeeta²

2009

B Soc Paran Mat

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Perhaps one of the major unsolved problem in Approximation Theory is : Whether or not every Chebyshev subset of a Hilbert space must be convex. Many partial answers to this problem are available in the literature. R.R. Phelps [Proc. Amer. Math. Soc. 8 (1957), 790-797] showed that a Chebyshev set in an inner product space (or in a strictly convex normed linear space) is convex if the associated metric projection is non-expansive. We extend this result to metric spaces.

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“…Particularly, he showed that if there is no point p ∈ S C (the complement of the set S), which is the center of a bitangent circle to S, S has to be convex. Klee (1949) and Phelps (1957) extended Motzkin's ideas to non Euclidean metrics.…”

Section: Introductionmentioning

confidence: 99%

Area-Based Medial Axis of Planar Curves

Niethammer

Betelú

Sapiro

et al. 2004

International Journal of Computer Vision

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A new definition of affine invariant medial axis of planar closed curves is introduced. A point belongs to the affine medial axis if and only if it is equidistant from at least two points of the curve, with the distance being a minimum and given by the areas between the curve and its corresponding chords. The medial axis is robust, eliminating the need for curve denoising. In a dynamical interpretation of this affine medial axis, the medial axis points are the affine shock positions of the affine erosion of the curve. We propose a simple method to compute the medial axis and give examples. We also demonstrate how to use this method to detect affine skew symmetry in real images.

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Convex sets and nearest points

Cited by 90 publications

References 8 publications

A topological approach to Best Approximation Theory

A topological approach to Best Approximation Theory

A note on the convexity of Chebyshev sets

Area-Based Medial Axis of Planar Curves

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