On the size of the sets of gradients of bump functions and starlike bodies on the Hilbert space

Daniel Azagra; Mar Jiménez-Sevilla

Bulletin de la Société Mathématique de France (2002)

  • Volume: 130, Issue: 3, page 337-347
  • ISSN: 0037-9484

Abstract

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We study the size of the sets of gradients of bump functions on the Hilbert space ℓ 2 , and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in ℓ 2 can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert space ℓ 2 can be uniformly approximated by C 1 smooth Lipschitz functions ψ so that the cones generated by the ranges of its derivatives ψ ' ( ℓ 2 ) have empty interior. This implies that there are C 1 smooth Lipschitz bumps in ℓ 2 so that the cones generated by their sets of gradients have empty interior. On the other hand, we construct C 1 -smooth bounded starlike bodies A ⊂ ℓ 2 , which approximate the unit ball, so that the cones generated by the hyperplanes which are tangent to A have empty interior as well. We also explain why this is the best answer to the above questions that one can expect.

How to cite

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Azagra, Daniel, and Jiménez-Sevilla, Mar. "On the size of the sets of gradients of bump functions and starlike bodies on the Hilbert space." Bulletin de la Société Mathématique de France 130.3 (2002): 337-347. <http://eudml.org/doc/272364>.

@article{Azagra2002,
abstract = {We study the size of the sets of gradients of bump functions on the Hilbert space $\ell _2$, and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in $\ell _2$ can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert space $\ell _2$ can be uniformly approximated by $C^1$ smooth Lipschitz functions $\psi $ so that the cones generated by the ranges of its derivatives $\psi ^\{\prime \}(\ell _2)$ have empty interior. This implies that there are $C^1$ smooth Lipschitz bumps in $\ell _2$ so that the cones generated by their sets of gradients have empty interior. On the other hand, we construct $C^1$-smooth bounded starlike bodies $A\subset \ell _2$, which approximate the unit ball, so that the cones generated by the hyperplanes which are tangent to $A$ have empty interior as well. We also explain why this is the best answer to the above questions that one can expect.},
author = {Azagra, Daniel, Jiménez-Sevilla, Mar},
journal = {Bulletin de la Société Mathématique de France},
keywords = {gradient; bump function; starlike body},
language = {eng},
number = {3},
pages = {337-347},
publisher = {Société mathématique de France},
title = {On the size of the sets of gradients of bump functions and starlike bodies on the Hilbert space},
url = {http://eudml.org/doc/272364},
volume = {130},
year = {2002},
}

TY - JOUR
AU - Azagra, Daniel
AU - Jiménez-Sevilla, Mar
TI - On the size of the sets of gradients of bump functions and starlike bodies on the Hilbert space
JO - Bulletin de la Société Mathématique de France
PY - 2002
PB - Société mathématique de France
VL - 130
IS - 3
SP - 337
EP - 347
AB - We study the size of the sets of gradients of bump functions on the Hilbert space $\ell _2$, and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in $\ell _2$ can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert space $\ell _2$ can be uniformly approximated by $C^1$ smooth Lipschitz functions $\psi $ so that the cones generated by the ranges of its derivatives $\psi ^{\prime }(\ell _2)$ have empty interior. This implies that there are $C^1$ smooth Lipschitz bumps in $\ell _2$ so that the cones generated by their sets of gradients have empty interior. On the other hand, we construct $C^1$-smooth bounded starlike bodies $A\subset \ell _2$, which approximate the unit ball, so that the cones generated by the hyperplanes which are tangent to $A$ have empty interior as well. We also explain why this is the best answer to the above questions that one can expect.
LA - eng
KW - gradient; bump function; starlike body
UR - http://eudml.org/doc/272364
ER -

References

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  1. [1] D. Azagra & R. Deville – « James’ theorem fails for starlike bodies », J. Funct. Anal.180 (2001), p. 328–346. Zbl0983.46016MR1814992
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  3. [3] J. Borwein, M. Fabian, I. Kortezov & P. Loewen – « The range of the gradient of a continuously differentiable bump », J. Nonlinear Convex Anal. 2 (2001), no. 1, p. 1–19. Zbl0993.46023MR1828155
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  6. [6] P. Hájek – « Smooth functions on c 0 », Israel J. Math.104 (1998), p. 17–27. Zbl0940.46023MR1622271
  7. [7] M. Jiménez-Sevilla & J. Moreno – « A note on norm attaining functionals », Proc. Amer. Math. Soc. 126 (1998), no. 3, p. 1989–1997. Zbl0894.46011MR1485482
  8. [8] J. Malý – « The Darboux property for gradients », Real Anal. Exchange 22 (1996/1997), p. 167–173. Zbl0879.26042MR1433604
  9. [9] R. Phelps – Convex functions, monotone operators and differentiability, Lecture Notes in Math., vol. 1364, Springer-Verlag, Berlin, 1989. Zbl0658.46035MR984602
  10. [10] S. Shkarin – « On Rolle’s theorem in infinite-dimensional Banach spaces », Mat. Zametki 51 (1992), no. 3, p. 128–136, English transl. Zbl0786.46044MR1172237

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