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

Daniel AzagraMar Jiménez-Sevilla — 2002

Bulletin de la Société Mathématique de France

We study the size of the sets of gradients of bump functions on the Hilbert space 2 , and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in 2 can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert space 2 can be uniformly approximated by C 1 smooth Lipschitz functions ψ so that the cones generated by the ranges of its derivatives ψ ' ( 2 ) have empty interior. This implies that there are C 1 smooth Lipschitz bumps...

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