"105673" . . "DE - Spolkov\u00E1 republika N\u011Bmecko" . "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 . . . . "Smallness of Singular Sets of Semiconvex Functions in Separable Banach Spaces" . . . . "[06063D6EA413]" . "Smallness of Singular Sets of Semiconvex Functions in Separable Banach Spaces"@en . . "Smallness of Singular Sets of Semiconvex Functions in Separable Banach Spaces"@en . . "Duda, Jakub" . "2" . . . . . . "P(GA201/09/0067), Z(MSM0021620839)" . . . . "20" . "000322348200015" . "26"^^ . "11320" . "1"^^ . "0944-6532" . "2"^^ . "Smallness of Singular Sets of Semiconvex Functions in Separable Banach Spaces" . "superreflexive space; DSC surface; Lipschitz surface; singular point of order k; singular set; Clarke subdifferential; Semiconvex function with general modulus"@en . . . . "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." . "RIV/00216208:11320/13:10189657!RIV14-GA0-11320___" . "Zaj\u00ED\u010Dek, Lud\u011Bk" . "RIV/00216208:11320/13:10189657" . . "Journal of Convex Analysis" . .