Lipschitz Mappings and Homeomorphisms

We study three sets of problems. All of them deal with Lipschitz mappings in Hilbert and Banach spaces. We examine how close almost isometries, that is, bilipschitz mappings with both Lipschitz constants almost one, are to linear mappings. We consider projections in Hilbert space, that is, nearest point mappings onto either closed subspaces, or, more generaly, closed convex subsets. We ask when iterations of such projections, drawn from a finite pool, converge. In the third set of problems we deal with nonexpansive mappings. We ask how the existence of fixed points of nonexpansive self-mappings of bounded closed convex sets is connected with reflexivity. We also study the possible extension of firmly nonexpansive mappings.

We study three sets of problems. All of them deal with Lipschitz mappings in Hilbert and Banach spaces. We examine how close almost isometries, that is, bilipschitz mappings with both Lipschitz constants almost one, are to linear mappings. We consider projections in Hilbert space, that is, nearest point mappings onto either closed subspaces, or, more generaly, closed convex subsets. We ask when iterations of such projections, drawn from a finite pool, converge. In the third set of problems we deal with nonexpansive mappings. We ask how the existence of fixed points of nonexpansive self-mappings of bounded closed convex sets is connected with reflexivity. We also study the possible extension of firmly nonexpansive mappings.