# 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

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topAzagra, 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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- [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] J. Malý – « The Darboux property for gradients », Real Anal. Exchange 22 (1996/1997), p. 167–173. Zbl0879.26042MR1433604
- [9] R. Phelps – Convex functions, monotone operators and differentiability, Lecture Notes in Math., vol. 1364, Springer-Verlag, Berlin, 1989. Zbl0658.46035MR984602
- [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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