Singular points of order k of Clarke regular and arbitrary functions

Luděk Zajíček

Commentationes Mathematicae Universitatis Carolinae (2012)

  • Volume: 53, Issue: 1, page 51-63
  • ISSN: 0010-2628

Abstract

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Let X be a separable Banach space and f a locally Lipschitz real function on X . For k ∈ ℕ , let Σ k ( f ) be the set of points x ∈ X , at which the Clarke subdifferential ∂ C f ( x ) is at least k -dimensional. It is well-known that if f is convex or semiconvex (semiconcave), then Σ k ( f ) can be covered by countably many Lipschitz surfaces of codimension k . We show that this result holds even for each Clarke regular function (and so also for each approximately convex function). Motivated by a resent result of A.D. Ioffe, we prove also two results on arbitrary functions, which work with Hadamard directional derivatives and can be considered as generalizations of our theorem on Σ k ( f ) of Clarke regular functions (since each of them easily implies this theorem).

How to cite

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Zajíček, Luděk. "Singular points of order $k$ of Clarke regular and arbitrary functions." Commentationes Mathematicae Universitatis Carolinae 53.1 (2012): 51-63. <http://eudml.org/doc/246741>.

@article{Zajíček2012,
abstract = {Let $X$ be a separable Banach space and $f$ a locally Lipschitz real function on $X$. For $k\in \mathbb \{N\}$, let $\Sigma _k(f)$ be the set of points $x\in X$, at which the Clarke subdifferential $\partial ^Cf(x)$ is at least $k$-dimensional. It is well-known that if $f$ is convex or semiconvex (semiconcave), then $\Sigma _k(f)$ can be covered by countably many Lipschitz surfaces of codimension $k$. We show that this result holds even for each Clarke regular function (and so also for each approximately convex function). Motivated by a resent result of A.D. Ioffe, we prove also two results on arbitrary functions, which work with Hadamard directional derivatives and can be considered as generalizations of our theorem on $\Sigma _k(f)$ of Clarke regular functions (since each of them easily implies this theorem).},
author = {Zajíček, Luděk},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Clarke regular functions; singularities; Hadamard derivative; Clarke regular function; singularity; Hadamard derivative},
language = {eng},
number = {1},
pages = {51-63},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Singular points of order $k$ of Clarke regular and arbitrary functions},
url = {http://eudml.org/doc/246741},
volume = {53},
year = {2012},
}

TY - JOUR
AU - Zajíček, Luděk
TI - Singular points of order $k$ of Clarke regular and arbitrary functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2012
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 53
IS - 1
SP - 51
EP - 63
AB - Let $X$ be a separable Banach space and $f$ a locally Lipschitz real function on $X$. For $k\in \mathbb {N}$, let $\Sigma _k(f)$ be the set of points $x\in X$, at which the Clarke subdifferential $\partial ^Cf(x)$ is at least $k$-dimensional. It is well-known that if $f$ is convex or semiconvex (semiconcave), then $\Sigma _k(f)$ can be covered by countably many Lipschitz surfaces of codimension $k$. We show that this result holds even for each Clarke regular function (and so also for each approximately convex function). Motivated by a resent result of A.D. Ioffe, we prove also two results on arbitrary functions, which work with Hadamard directional derivatives and can be considered as generalizations of our theorem on $\Sigma _k(f)$ of Clarke regular functions (since each of them easily implies this theorem).
LA - eng
KW - Clarke regular functions; singularities; Hadamard derivative; Clarke regular function; singularity; Hadamard derivative
UR - http://eudml.org/doc/246741
ER -

References

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  12. Zajíček L., On the differentiation of convex functions in finite and infinite dimensional spaces, Czechoslovak Math. J. 29 (1979), 340–348. MR0536060
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