We prove that the regular n × n square grid of points in the integer lattice Z 2 cannot be recovered from an arbitrary n 2 -element subset of Z 2 via a mapping with prescribed Lipschitz constant (independent of n). This answers negatively a question of Feige from 2002. Our resolution of Feige's question takes place largely in a continuous setting and is based on some new results for Lipschitz mappings falling into two broad areas of interest, which we study independently. Firstly the present work contains a detailed investigation of Lipschitz regular mappings on Euclidean spaces, with emphasis on their bilipschitz decomposability in a sense comparable to that of the well known result of Jones. Secondly, we build on work of Burago and Kleiner and McMullen on non-realisable densities. We verify the existence, and further prevalence, of strongly non-realisable densities inside spaces of continuous functions.

We show that every finite-dimensional Euclidean space contains compact universal differentiability sets of upper Minkowski dimension one. In other words, there are compact sets S of upper Minkowski dimension one such that every Lipschitz function defined on the whole space is differentiable inside S. Such sets are constructed explicitly.

Abstract. We consider the space of non-expansive mappings on a bounded, closed and convex subset of a Banach space equipped with the metric of uniform convergence.We show that the set of strict contractions is a σ-porous subset. If the underlying Banach space is separable, we exhibit a σ-porous subset of the space of non-expansive mappings outside of which all mappings attain the maximal Lipschitz constant one at typical points of their domain.

Abstract. We consider a large class of geodesic metric spaces, including Banach spaces, hyperbolic spaces and geodesic CAT(κ)-spaces, and investigate the space of nonexpansive mappings on either a convex or a star-shaped subset in these settings. We prove that the strict contractions form a negligible subset of this space in the sense that they form a σ-porous subset. For certain separable and complete metric spaces we show that a generic nonexpansive mapping has Lipschitz constant one at typical points of its domain. These results contain the case of nonexpansive self-mappings and the case of nonexpansive set-valued mappings as particular cases. Mathematics Subject Classification (2010). 47H09, 47H04, 54E52

In 1998 Burago and Kleiner and (independently) McMullen gave examples of separated nets in Euclidean space which are non-bilipschitz equivalent to the integer lattice. We study weaker notions of equivalence of separated nets and demonstrate that such notions also give rise to distinct equivalence classes. Put differently, we find occurrences of particularly strong divergence of separated nets from the integer lattice. Our approach generalises that of Burago and Kleiner and McMullen which takes place largely in a continuous setting. Existence of irregular separated nets is verified via the existence of non-realisable density functions ρ : [0, 1] d → (0, ∞). In the present work we obtain stronger types of non-realisable densities.