 
Summary: ON THE SIZE OF THE SETS OF GRADIENTS OF BUMP
FUNCTIONS AND STARLIKE BODIES ON THE HILBERT
SPACE
DANIEL AZAGRA AND MAR JIM´ENEZSEVILLA
Abstract. 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 C1
smooth Lipschitz functions so that
the cones generated by the ranges of its derivatives ( 2) have empty interior.
This implies that there are C1
smooth Lipschitz bumps in 2 so that the cones
generated by their sets of gradients have empty interior. On the other hand, we
construct C1
smooth bounded starlike bodies A 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.
R´ESUM´E. On ´etudie la taille de l'ensemble de valeurs du gradient d'une fonction lisse, non
