RT Journal Article
T1 On the size of the sets of gradients of Bump functions and starlike bodies on the Hilbert space
A1 Azagra Rueda, Daniel
A1 Jiménez Sevilla, María Del Mar
AB 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.
PB Société Mathématique de France
SN 0037-9484
YR 2002
FD 2002
LK https://hdl.handle.net/20.500.14352/57098
UL https://hdl.handle.net/20.500.14352/57098
LA eng
NO DGICYT
DS Docta Complutense
RD 7 abr 2025