Smallness of Singular Sets of Semiconvex Functions in Separable Banach Spaces

Result code in 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>
Result on the web
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DOI - Digital Object Identifier
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Result language
angličtina
Original language name
Smallness of Singular Sets of Semiconvex Functions in Separable Banach Spaces
Original language 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 count
Czech name
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Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Project
<a href="/en/project/GA201%2F09%2F0067" target="_blank" >GA201/09/0067: Theory of real functions and descriptive set theory II</a><br>
Continuities
Z - Vyzkumny zamer (s odkazem do CEZ)

Publication year
2013
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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