Lipschitz Mappings and Homeomorphisms

We study three sets of problems. All of them deal with Lipschitz mappings in Hilbert and Banach spaces. We examine when can be an almost isometry, defined on finite subset of the unit ball of a Hilbert space, extended to bilipschitz almost isometry of the entire ball. 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 approximate contractive retracts.

We have studied four sets of problems. Their common themes are Lipschitz mappings and fixed points. We consider products of orthogonal projections onto finitely many closed subspaces of Hilbert space. We estimate their rate of convergence when the subspaces have finite dimension, or codimension. The proof is based on an involved construction of a Tietze-like potential, and gives a new approach to the study of iterative projection methods. Every contraction from a bounded subset of a Euclidean space into that space can be extended to the entire space to be a contraction again. We show that there are extension operators continuous in the supremum norm. The multivalued extension operator is lower semicontinuous. As a corollary, Lipschitz isometries are residual in the contractions. We study the resolvents of coaccretive operators in the Hilbert ball, with special emphasis on the asymptotic behavior of their compositions and metric convex combinations. We establish both weak and strong convergence results. We prove the existence of zone diagrams with respect to finitely many pairwise disjoint compact sites contained in a compact and convex subset of a uniformly convex normed space. We obtain the continuity of the Dom mapping as well as interesting and apparently new properties of Voronoi cells. Infinite products of projections find application in computer tomography while zone diagrams originate in the design of integrated circuits.