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

Abstract

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 ψ so that the cones generated by the ranges of its derivatives ψ ' (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⊂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.