Weakly Convex Sets and Their Properties

Ivanov, Grigorii Evgen'evich

doi:10.1007/s11006-006-0005-y

Cited by 16 publications

(10 citation statements)

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“…The next Lemma is a generalization of Lemma 1.9.1 from G.E. Ivanov's monograph [5], in which an analogous result was obtained for Hilbert spaces.…”

Section: Holdsmentioning

confidence: 59%

“…Some notes about the history of this problem one could find in paper [3]. Perhaps, the most complete description of the proximally smooth sets and their properties in a Hilbert space can be found in [5].…”

Section: Introductionmentioning

confidence: 99%

“…The problem of characterization of proximally smooth sets in Hilbert spaces through the monotony properties of the normal cone turns out to be very important for applications. Thus, Theorem 1.9.1 of monograph [5] may be reformulated in the following way (See also [3], Cor.…”

Section: Introductionmentioning

confidence: 99%

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Hypomonotonicity of the normal cone and proximal smoothness

Ivanov¹

2015

Preprint

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In this paper we study the properties of the normal cone to the proximally smooth set. We give the complete characterization of a proximally smooth set through the monotony properties of its normal cone in an arbitrary uniformly convex and uniformly smooth Banach space. We give the exact bounds for right-hand side in the monotonicity inequality for normal cone in terms of the moduli of smoothness and convexity of a Banach space.

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“…The next Lemma is a generalization of Lemma 1.9.1 from G.E. Ivanov's monograph [5], in which an analogous result was obtained for Hilbert spaces.…”

Section: Holdsmentioning

confidence: 59%

Section: Introductionmentioning

confidence: 99%

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Hypomonotonicity of the normal cone and proximal smoothness

Ivanov¹

2015

Preprint

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“…(3) α < 0, then T is a hypomonotone operator. For example, the subdifferential of a prox-regular function on a Hilbert space is a hypomonotone operator (see [19]). Inequality ( 16) is often called the variational inequality.…”

Section: Ifmentioning

confidence: 99%

New moduli for Banach spaces

Ivanov¹,

Martini²

2017

Ann. Funct. Anal.

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Modifying the moduli of supporting convexity and supporting smoothness, we introduce new moduli for Banach spaces which occur, e.g., as lengths of catheti of right-angled triangles (defined via so-called quasi-orthogonality). These triangles have two boundary points of the unit ball of a Banach space as endpoints of their hypotenuse, and their third vertex lies in a supporting hyperplane of one of the two other vertices. Among other things it is our goal to quantify via such triangles the local deviation of the unit sphere from its supporting hyperplanes. We prove respective Day-Nordlander type results, involving generalizations of the modulus of convexity and the modulus of Banaś.Mathematics Subject Classification (2010): 46B07, 46B20, 52A10, 52A20, 52A21

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“…xS  , значения операторов Минковского совпадают соответственно с суммой и разностью Минковского: [12], [13], [14], [15]. Расчет геометрической разности был в основном необходим при определении адекватности условий в примерах [15]. Сегодня приближенное вычисление суммы и разности Минковского играет важную роль в решении практических задач с помощью дифференциальных игр.…”

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