About: Smallness of Singular Sets of Semiconvex Functions in Separable Banach Spaces

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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. (en)
Title	Smallness of Singular Sets of Semiconvex Functions in Separable Banach Spaces Smallness of Singular Sets of Semiconvex Functions in Separable Banach Spaces (en)
skos:prefLabel	Smallness of Singular Sets of Semiconvex Functions in Separable Banach Spaces Smallness of Singular Sets of Semiconvex Functions in Separable Banach Spaces (en)
skos:notation	RIV/00216208:11320/13:10189657!RIV14-GA0-11320___
http://linked.open...avai/riv/aktivita	P Z
http://linked.open...avai/riv/aktivity	P(GA201/09/0067), Z(MSM0021620839)
http://linked.open...iv/cisloPeriodika	2
http://linked.open...vai/riv/dodaniDat	2014
http://linked.open...aciTvurceVysledku	Zajíček, Luděk
http://linked.open.../riv/druhVysledku	J - Článek v odborném periodiku
http://linked.open...iv/duvernostUdaju	S - Úplné a pravdivé údaje nepodléhající ochraně podle zvláštních právních předpisů
http://linked.open...titaPredkladatele	Univerzita Karlova v Praze / Matematicko-fyzikální fakulta
http://linked.open...dnocenehoVysledku	105673
http://linked.open...ai/riv/idVysledku	RIV/00216208:11320/13:10189657
http://linked.open...riv/jazykVysledku	eng - angličtina
http://linked.open.../riv/klicovaSlova	superreflexive space; DSC surface; Lipschitz surface; singular point of order k; singular set; Clarke subdifferential; Semiconvex function with general modulus (en)
http://linked.open.../riv/klicoveSlovo	DSC surface Lipschitz surface Semiconvex function with general modulus singular point of order k singular set superreflexive space Clarke subdifferential
http://linked.open...odStatuVydavatele	DE - Spolková republika Německo
http://linked.open...ontrolniKodProRIV	[06063D6EA413]
http://linked.open...i/riv/nazevZdroje	Journal of Convex Analysis
http://linked.open...in/vavai/riv/obor	BA
http://linked.open...ichTvurcuVysledku	1 (xsd:int)
http://linked.open...cetTvurcuVysledku	2 (xsd:int)
http://linked.open...vavai/riv/projekt	Theory of real functions and descriptive set theory II
http://linked.open...UplatneniVysledku	2013
http://linked.open...v/svazekPeriodika	20
http://linked.open...iv/tvurceVysledku	Duda, Jakub Zajíček, Luděk
http://linked.open...ain/vavai/riv/wos	000322348200015
http://linked.open...n/vavai/riv/zamer	Metody moderní matematiky a jejich aplikace
issn	0944-6532
number of pages	26 (xsd:int)
http://localhost/t...ganizacniJednotka	11320

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