2023-12-01T03:39:43Zhttps://docta.ucm.es/rest/oai/requestoai:docta.ucm.es:20.500.14352/570982023-08-26T18:54:51Zcom_20.500.14352_14col_20.500.14352_15
On the size of the sets of gradients of Bump functions and starlike bodies on the Hilbert space
Azagra Rueda, Daniel
Jiménez Sevilla, María del Mar
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 psi so that the cones generated by the ranges of its derivatives psi'(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 subset of 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.
2023-06-20T16:48:42Z
2023-06-20T16:48:42Z
2023-06-20T16:48:42Z
2002
journal article
https://hdl.handle.net/20.500.14352/57098
0037-9484
http://smf4.emath.fr/Publications/Bulletin/
eng
PB 96-0607
BFM 2000-0609
open access
Société Mathématique de France