Displaying similar documents to “A d.c. C 1 function need not be difference of convex C 1 functions”

The distance between subdifferentials in the terms of functions

Libor Veselý (1993)

Commentationes Mathematicae Universitatis Carolinae

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For convex continuous functions f , g defined respectively in neighborhoods of points x , y in a normed linear space, a formula for the distance between ∂ f ( x ) and ∂ g ( y ) in terms of f , g (i.eẇithout using the dual) is proved. Some corollaries, like a new characterization of the subdifferential of a continuous convex function at a point, are given. This, together with a theorem from [4], implies a sufficient condition for a family of continuous convex functions on a barrelled normed linear space to be locally...

A "hidden" characterization of approximatively polyhedral convex sets in Banach spaces

Taras Banakh, Ivan Hetman (2012)

Studia Mathematica

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A closed convex subset C of a Banach space X is called approximatively polyhedral if for each ε > 0 there is a polyhedral (= intersection of finitely many closed half-spaces) convex set P ⊂ X at Hausdorff distance < ε from C. We characterize approximatively polyhedral convex sets in Banach spaces and apply the characterization to show that a connected component of the space C o n v ( X ) of closed convex subsets of X endowed with the Hausdorff metric is separable if and only if contains a...

Generalized characterization of the convex envelope of a function

Fethi Kadhi (2002)

RAIRO - Operations Research - Recherche Opérationnelle

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We investigate the minima of functionals of the form ∫ [ a , b ] g ( u ˙ ( s ) ) d s where g is strictly convex. The admissible functions u : [ a , b ] → ℝ are not necessarily convex and satisfy u ≤ f on [ a , b ] , u ( a ) = f ( a ) , u ( b ) = f ( b ) , f is a fixed function on [ a , b ] . We show that the minimum is attained by f ¯ , the convex envelope of f .

On Bárány's theorems of Carathéodory and Helly type

Ehrhard Behrends (2000)

Studia Mathematica

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The paper begins with a self-contained and short development of Bárány’s theorems of Carathéodory and Helly type in finite-dimensional spaces together with some new variants. In the second half the possible generalizations of these results to arbitrary Banach spaces are investigated. The Carathéodory-Bárány theorem has a counterpart in arbitrary dimensions under suitable uniform compactness or uniform boundedness conditions. The proper generalization of the Helly-Bárány theorem reads...

r –convex transformability in nonlinear programming problems

Elżbieta Galewska, Marek Galewski (2005)

Commentationes Mathematicae Universitatis Carolinae

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We show that for r -convex transformable nonlinear programming problems the Karush-Kuhn-Tucker necessary optimality conditions are also sufficient and we provide a method of solving such problems with the aid of associated r -convex ones.