The support functionals of a convex set

Bishop, Errett; Phelps, R. R.

doi:10.1090/pspum/007/0154092

Cited by 201 publications

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“…Any such functional is said to be a support functional of C and the set of all support points is denoted by supp C. It is known [1] that the support points of C are always dense in bdry C, the boundary of C, and that the support functional of C are norm dense among those which are bounded above on C. A corollary of the methods used for these results is the fact that C is always the intersection of all those closed half-spaces which are defined by support functionals [1,Corollary 2]. This result is trivial, of course, if C has nonempty interior, since every boundary point of C is a support point.…”

Section: Removable Sets Of Support Points Of Convex Sets In Banach Spmentioning

confidence: 99%

Removable Sets of Support Points of Convex Sets in Banach Spaces

Phelps¹

1987

Proceedings of the American Mathematical Society

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ABSTRACT. A corollary of the Bishop-Phelps theorem is that a closed convex subset C of a Banach space can always be represented as the intersection of its supporting closed half-spaces. In this paper an investigation is made of those subsets S of C such that C is the intersection of those closed half-spaces which support it at points of C\S.This will be true for sets S which are "small" relative to C, where smallness can be measured in terms of dimension, density character, or tr-compactness.Suppose that C is a nonempty closed convex subset of a Banach space E. A point x G C is called a support point of C if there exists a nonzero functional / G E* which attains its supremum on C at x. Any such functional is said to be a support functional of C and the set of all support points is denoted by supp C. It is known [1] that the support points of C are always dense in bdry C, the boundary of C, and that the support functional of C are norm dense among those which are bounded above on C. A corollary of the methods used for these results is the fact that C is always the intersection of all those closed half-spaces which are defined by support functionals [1, Corollary 2]. This result is trivial, of course, if C has nonempty interior, since every boundary point of C is a support point. In this case, in fact, it is easily seen that C can be represented as the intersection of those halfspaces which support it at the points of D for any dense subset D of bdry C (see part (iv) of Theorem 1, below). This fact has played a key role in characterizing those generators of Co-semigroups of operators which leave invariant a given closed convex set with interior [2,3]. In considering the extension of his work [2] to more general convex sets, K. N. Boyadzhiev raised the question (in a letter to the author) of whether one could express C as the intersection of those closed half-spaces which support C at some proper subset of supp C. The purpose of this note is to give some answers to this question.A little thought shows that one has to use some care in deleting subsets of supp C. For instance, if C is a line segment, then a point x of F\C on the line determined by C can be separated from C only by those support functionals which attain their maximum on C at the endpoint nearest x; that is, one cannot remove that endpoint from supp C and still separate x from C by a support functional. (If the dimension Of E is at least two, then every point of C is a support point, so supp C minus a single point is still dense in bdry C.) If C is infinite dimensional, then one can remove a finite subset S of supp C; in fact, as we show below, one can remove a finite dimensional subset and still obtain C as the intersection of half-spaces which

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Section: Removable Sets Of Support Points Of Convex Sets In Banach Spmentioning

confidence: 99%

Removable Sets of Support Points of Convex Sets in Banach Spaces

Phelps¹

1987

Proceedings of the American Mathematical Society

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“…while A is said to be locally accretive if (1) holds for all X > 0 when v is sufficiently near w. The nonlinear accretive operators were introduced independently in 1967 by F. E. Browder [3] and T. Kato [9], and they observed that A is accretive if and only if j(Aw -Ay) > 0 for some 7 E X* satisfying \\j\\ = \\w -y\\ and j(w -y) = || w -v||2. An early fundamental result in the theory of accretive operators, due to Browder [4], states that the initial value problem % + Au = 0,…”

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confidence: 99%

“…The above theorem is essentially equivalent to a theorem of I. Ekeland [8], which is in turn an abstraction of a lemma due to Bishop and Phelps [1]. Since its appearance in 1976, this theorem has found numerous applications in varied areas (see [2] or [10] for example).…”

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confidence: 99%

An Elementary Proof of Surjectivity for a Class of Accretive Operators

Ray¹

1979

Proceedings of the American Mathematical Society

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Abstract. An operator A defined on a real Banach space X is said to be locally accretive if, for each A > 0, x e X and each v near x, \\x -v|| < ||jc -y + \(Ax -Ay)\\. It is shown that if A : X -» X is locally accretive and locally Lipschitzian then (/ + A)(X) = X.Let X be a real Banach space. A nonlinear operator A : X -» X is said to be accretive if, for each X > 0 and for each w, y E X,while A is said to be locally accretive if (1) is solvable when A is locally Lipschitzian and accretive on X, a result which was subsequently generalized by R. H. Martin [12] to the continuous accretive operators. The existence theory for equation (2) [12] showed that if A is locally Lipschitzian (in Martin's theorem only continuity is required) and accretive, then A is m-accretive, i.e., (/ + A)(X) = X. Recently M. Crandall and A. Pazy [7] have given a proof of Martin's result which does not rely on solvability of (2), but which remains rather complex, being based on an intricate iteration scheme. It is our purpose in this note to present an elementary and straightforward proof of Browder's theorem for locally accretive operators.

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“…By Theorem 5, cl(f(X)) is convex. By hypothesis, f(X) is closed in X so that ƒ (X) is a closed convex subset of Y If/ (X) # X then by a simple variant of the Bishop-Phelps Theorem [1], there exists a support point of/(X) in X i-e. a nonzero y* in Y* and an element y = ƒ (u) inf(X) such that (y*, y) = sup xeX (y*,f(x)).…”

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confidence: 99%

Normal solvability and 𝜙-accretive mappings of Banach spaces

Browder¹

1972

Bull. Amer. Math. Soc.

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The support functionals of a convex set

Cited by 201 publications

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Removable Sets of Support Points of Convex Sets in Banach Spaces

Removable Sets of Support Points of Convex Sets in Banach Spaces

An Elementary Proof of Surjectivity for a Class of Accretive Operators

Normal solvability and 𝜙-accretive mappings of Banach spaces

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