Azagra Rueda, Daniel and Jimenez Sevilla, Maria del Mar (2002) 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). pp. 337-347. ISSN 0037-9484
Official URL: http://smf4.emath.fr/Publications/Bulletin/
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.
|Uncontrolled Keywords:||Gradient; Bump function; Starlike body; Rolles Theorem|
|Subjects:||Sciences > Mathematics > Functional analysis and Operator theory|
|Deposited On:||06 Sep 2011 09:28|
|Last Modified:||06 Sep 2011 09:28|
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