Diffeomorphisms between spheres and hyperplanes in infinite-dimensional Banach spaces
On diffeomorphisms deleting weak compacta in Banach spaces
We prove that if X is an infinite-dimensional Banach space with smooth partitions of unity then X and X∖ K are diffeomorphic for every weakly compact set K ⊂ X.
Starlike bodies and deleting diffeomorphisms in Banach spaces.
Inf-convolution and regularization of convex functions on Riemannian manifolds of nonpositive curvature.
We show how an operation of inf-convolution can be used to approximate convex functions with C smooth convex functions on Riemannian manifolds with nonpositive curvature (in a manner that not only is explicit but also preserves some other properties of the original functions, such as ordering, symmetries, infima and sets of minimizers), and we give some applications.
On the size of the sets of gradients of bump functions and starlike bodies on the Hilbert space
We study the size of the sets of gradients of bump functions on the Hilbert space , and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert space can be uniformly approximated by smooth Lipschitz functions so that the cones generated by the ranges of its derivatives have empty interior. This implies that there are smooth Lipschitz bumps...
Geometrical and topological properties of bumps and starlike bodies in Banach spaces.
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