The distance between subdifferentials in the terms of functions

Libor Veselý

Commentationes Mathematicae Universitatis Carolinae (1993)

  • Volume: 34, Issue: 3, page 419-424
  • ISSN: 0010-2628

Abstract

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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 uniformly Lipschitz.

How to cite

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Veselý, Libor. "The distance between subdifferentials in the terms of functions." Commentationes Mathematicae Universitatis Carolinae 34.3 (1993): 419-424. <http://eudml.org/doc/247467>.

@article{Veselý1993,
abstract = {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 $\partial f(x)$ and $\partial g(y)$ 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 uniformly Lipschitz.},
author = {Veselý, Libor},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {convex analysis; subdifferentials of convex functions; barrelled normed linear spaces; subdifferential of convex functions; distance; local uniform Lipschitz property},
language = {eng},
number = {3},
pages = {419-424},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The distance between subdifferentials in the terms of functions},
url = {http://eudml.org/doc/247467},
volume = {34},
year = {1993},
}

TY - JOUR
AU - Veselý, Libor
TI - The distance between subdifferentials in the terms of functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1993
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 34
IS - 3
SP - 419
EP - 424
AB - 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 $\partial f(x)$ and $\partial g(y)$ 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 uniformly Lipschitz.
LA - eng
KW - convex analysis; subdifferentials of convex functions; barrelled normed linear spaces; subdifferential of convex functions; distance; local uniform Lipschitz property
UR - http://eudml.org/doc/247467
ER -

References

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  1. Giles J.R., Convex Analysis with Application in Differentiation of Convex Functions, Research Notes in Mathematics, Vol. 58, Pitman, Boston-London-Melbourne, 1982. MR0650456
  2. Phelps R.R., Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics, Vol. 1364, Springer-Verlag, Berlin-New York-Heidelberg, 1989. Zbl0921.46039MR0984602
  3. Roberts A.W., Varberg D.E., Convex Functions, Academic Press, New York-San Francisco- London, 1973. Zbl0289.26012MR0442824
  4. Veselý L., Local uniform boundedness principle for families of ε -monotone operators, to appear. MR1326107

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