On sets of non-differentiability of Lipschitz and convex functions

Luděk Zajíček

Mathematica Bohemica (2007)

  • Volume: 132, Issue: 1, page 75-85
  • ISSN: 0862-7959

Abstract

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We observe that each set from the system 𝒜 ˜ (or even 𝒞 ˜ ) is Γ -null; consequently, the version of Rademacher’s theorem (on Gâteaux differentiability of Lipschitz functions on separable Banach spaces) proved by D. Preiss and the author is stronger than that proved by D. Preiss and J. Lindenstrauss. Further, we show that the set of non-differentiability points of a convex function on n is σ -strongly lower porous. A discussion concerning sets of Fréchet non-differentiability points of continuous convex functions on a separable Hilbert space is also presented.

How to cite

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Zajíček, Luděk. "On sets of non-differentiability of Lipschitz and convex functions." Mathematica Bohemica 132.1 (2007): 75-85. <http://eudml.org/doc/250264>.

@article{Zajíček2007,
abstract = {We observe that each set from the system $\widetilde\{\mathcal \{A\}\}$ (or even $\widetilde\{\mathcal \{C\}\}$) is $\Gamma $-null; consequently, the version of Rademacher’s theorem (on Gâteaux differentiability of Lipschitz functions on separable Banach spaces) proved by D. Preiss and the author is stronger than that proved by D. Preiss and J. Lindenstrauss. Further, we show that the set of non-differentiability points of a convex function on $\{\mathbb \{R\}\}^n$ is $\sigma $-strongly lower porous. A discussion concerning sets of Fréchet non-differentiability points of continuous convex functions on a separable Hilbert space is also presented.},
author = {Zajíček, Luděk},
journal = {Mathematica Bohemica},
keywords = {Lipschitz function; convex function; Gâteaux differentiability; Fréchet differentiability; $\Gamma $-null sets; ball small sets; $\delta $-convex surfaces; strong porosity; Gâteaux differentiability; Fréchet differentiability; -null sets; ball small sets; -convex surfaces; strong porosity},
language = {eng},
number = {1},
pages = {75-85},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On sets of non-differentiability of Lipschitz and convex functions},
url = {http://eudml.org/doc/250264},
volume = {132},
year = {2007},
}

TY - JOUR
AU - Zajíček, Luděk
TI - On sets of non-differentiability of Lipschitz and convex functions
JO - Mathematica Bohemica
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 132
IS - 1
SP - 75
EP - 85
AB - We observe that each set from the system $\widetilde{\mathcal {A}}$ (or even $\widetilde{\mathcal {C}}$) is $\Gamma $-null; consequently, the version of Rademacher’s theorem (on Gâteaux differentiability of Lipschitz functions on separable Banach spaces) proved by D. Preiss and the author is stronger than that proved by D. Preiss and J. Lindenstrauss. Further, we show that the set of non-differentiability points of a convex function on ${\mathbb {R}}^n$ is $\sigma $-strongly lower porous. A discussion concerning sets of Fréchet non-differentiability points of continuous convex functions on a separable Hilbert space is also presented.
LA - eng
KW - Lipschitz function; convex function; Gâteaux differentiability; Fréchet differentiability; $\Gamma $-null sets; ball small sets; $\delta $-convex surfaces; strong porosity; Gâteaux differentiability; Fréchet differentiability; -null sets; ball small sets; -convex surfaces; strong porosity
UR - http://eudml.org/doc/250264
ER -

References

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