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Statements

Subject Item
n2:RIV%2F00216208%3A11320%2F13%3A10189657%21RIV14-GA0-11320___
rdf:type
skos:Concept n4:Vysledek
dcterms:description
Let X be a separable superreflexive Banach space and f be a semiconvex function (with a general modulus) on X. For k epsilon N, let Sigma(k)(f) be the set of points x epsilon X, at which the Clarke subdifferential partial derivative f(x) is at least k-dimensional. Note that Sigma(1)(f) is the set of all points at which f is not Gateaux differentiable. Then Sigma(k)(f) can be covered by countably many Lipschitz surfaces of codimension k which are described by functions, which are differences of two semiconvex functions. If X is separable and superreflexive Banach space which admits an equivalent norm with modulus of smoothness of power type 2 (e.g., if X is a Hilbert space or X = L-p(mu) with 2 {= p), we give, for a fixed modulus w and k epsilon N, a complete characterization of those A subset of X, for which there exists a function f on X which is semiconvex on X with modulus w and A subset of Sigma(k)(f). Namely, A subset of X has this property if and only if A can be covered by countably many Lipschitz surfaces S-n f codimension k which are described by functions, which are differences of two Lipschitz semiconvex functions with modulus C(n)w. Let X be a separable superreflexive Banach space and f be a semiconvex function (with a general modulus) on X. For k epsilon N, let Sigma(k)(f) be the set of points x epsilon X, at which the Clarke subdifferential partial derivative f(x) is at least k-dimensional. Note that Sigma(1)(f) is the set of all points at which f is not Gateaux differentiable. Then Sigma(k)(f) can be covered by countably many Lipschitz surfaces of codimension k which are described by functions, which are differences of two semiconvex functions. If X is separable and superreflexive Banach space which admits an equivalent norm with modulus of smoothness of power type 2 (e.g., if X is a Hilbert space or X = L-p(mu) with 2 {= p), we give, for a fixed modulus w and k epsilon N, a complete characterization of those A subset of X, for which there exists a function f on X which is semiconvex on X with modulus w and A subset of Sigma(k)(f). Namely, A subset of X has this property if and only if A can be covered by countably many Lipschitz surfaces S-n f codimension k which are described by functions, which are differences of two Lipschitz semiconvex functions with modulus C(n)w.
dcterms:title
Smallness of Singular Sets of Semiconvex Functions in Separable Banach Spaces Smallness of Singular Sets of Semiconvex Functions in Separable Banach Spaces
skos:prefLabel
Smallness of Singular Sets of Semiconvex Functions in Separable Banach Spaces Smallness of Singular Sets of Semiconvex Functions in Separable Banach Spaces
skos:notation
RIV/00216208:11320/13:10189657!RIV14-GA0-11320___
n4:predkladatel
n5:orjk%3A11320
n3:aktivita
n13:Z n13:P
n3:aktivity
P(GA201/09/0067), Z(MSM0021620839)
n3:cisloPeriodika
2
n3:dodaniDat
n8:2014
n3:domaciTvurceVysledku
n18:7357362
n3:druhVysledku
n14:J
n3:duvernostUdaju
n9:S
n3:entitaPredkladatele
n12:predkladatel
n3:idSjednocenehoVysledku
105673
n3:idVysledku
RIV/00216208:11320/13:10189657
n3:jazykVysledku
n19:eng
n3:klicovaSlova
superreflexive space; DSC surface; Lipschitz surface; singular point of order k; singular set; Clarke subdifferential; Semiconvex function with general modulus
n3:klicoveSlovo
n7:Lipschitz%20surface n7:superreflexive%20space n7:Clarke%20subdifferential n7:singular%20point%20of%20order%20k n7:singular%20set n7:DSC%20surface n7:Semiconvex%20function%20with%20general%20modulus
n3:kodStatuVydavatele
DE - Spolková republika Německo
n3:kontrolniKodProRIV
[06063D6EA413]
n3:nazevZdroje
Journal of Convex Analysis
n3:obor
n20:BA
n3:pocetDomacichTvurcuVysledku
1
n3:pocetTvurcuVysledku
2
n3:projekt
n15:GA201%2F09%2F0067
n3:rokUplatneniVysledku
n8:2013
n3:svazekPeriodika
20
n3:tvurceVysledku
Duda, Jakub Zajíček, Luděk
n3:wos
000322348200015
n3:zamer
n11:MSM0021620839
s:issn
0944-6532
s:numberOfPages
26
n17:organizacniJednotka
11320