Smallness of Singular Sets of Semiconvex Functions in Separable Banach Spaces

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F13%3A10189657" target="_blank" >RIV/00216208:11320/13:10189657 - isvavai.cz</a>

Výsledek na webu

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DOI - Digital Object Identifier

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Jazyk výsledku

angličtina

Název v původním jazyce

Smallness of Singular Sets of Semiconvex Functions in Separable Banach Spaces

Popis výsledku v původním jazyce

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 count

Název v anglickém jazyce

Smallness of Singular Sets of Semiconvex Functions in Separable Banach Spaces

Popis výsledku anglicky

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 count

Druh

J_x - Nezařazeno - Článek v odborném periodiku (Jimp, Jsc a Jost)

CEP obor

BA - Obecná matematika

OECD FORD obor

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Projekt

<a href="/cs/project/GA201%2F09%2F0067" target="_blank" >GA201/09/0067: Teorie reálných funkcí a deskriptivní teorie množin II</a><br>

Návaznosti

Z - Vyzkumny zamer (s odkazem do CEZ)

Rok uplatnění

2013

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of Convex Analysis

ISSN

0944-6532

e-ISSN

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Svazek periodika

20

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

26

Strana od-do

573-598

Kód UT WoS článku

000322348200015

EID výsledku v databázi Scopus

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