Skip to main content

Nonlinear quotients and geometry of banach spaces

Final Activity Report Summary - NONLINEAR QUOTIENTS (Nonlinear Quotients and Geometry of Banach Spaces)

For my purposes it is important to understand the geometric structure of various deformations and transformations. (An obvious example, that is mildly connected with this research, is in elasticity.) Various classes of such deformations that correspond to different geometric situations were studied in the framework of this project, for example, Lipschitz functions, Lipschitz quotients, uniform co-Lipschitz functions and ball non collapsing functions. The key problem of the study of the local behaviour is whether deformations look like affine ones close to some points, or even close to many points; the relevant classes of exceptional points studied in this project were, for example, cone null sets, regularly cone null sets and sigma-porous sets.

The main results include an invariance theorem for cone null and regularly cone null sets under bi-Lipschitz isomorphisms, a construction of unexpectedly large sigma-porous sets in metric spaces admitting certain Lipschitz quotients and a study of a non-linear version of the Besicovitch-Federer projection theorem. These results are relevant especially for future development of geometric non-linear functional analysis and neighbouring fields.