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We prove that every $L$-bilipschitz mapping $\mathbb{Z}^2\to\mathbb{R}^2$ can be extended to a $C(L)$-bilipschitz mapping $\mathbb{R}^2\to\mathbb{R}^2$ and provide an upper bound for $C(L)$. Moreover, we extend the result to every separated net in $\mathbb{R}^2$ instead of $\mathbb{Z}^2$. Along the way, we develop a set of tools for bilipschitz extensions of mappings between subsets of Euclidean spaces.

We study the extremality of nonexpansive mappings on a nonempty bounded closed and convex subset of a normed space (therein specific Banach spaces). We show that surjective isometries are extremal in this sense for many Banach spaces, including Banach spaces with the Radon-Nikodym property and all $C(K)$-spaces for compact Hausdorff $K$. We also conclude that the typical, in the sense of Baire category, nonexpansive mapping is close to being extremal., Comment: 23 pages, minor changes to the previous version

We prove that a typical Lipschitz mapping between any two Banach spaces is non-differentiable at typical points of any given subset of its domain in the most extreme form. This is a new result even for Lipschitz mappings between Euclidean spaces.

For any non-trivial convex and bounded subset $C$ of a Banach space, we show that outside of a $\sigma$-porous subset of the space of non-expansive mappings $C\to C$, all mappings have the maximal Lipschitz constant one witnessed locally at typical points of $C$. This extends a result of Bargetz and the author from separable Banach spaces to all Banach spaces and the proof given is completely independent. We further establish a fine relationship between the classes of exceptional sets involved in this statement, captured by the hierarchy of notions of $\phi$-porosity., Comment: A few corrections and improvements made. To appear in Israel Journal of Mathematics

We introduce a hierachy of equivalence relations on the set of separated nets of a given Euclidean space, indexed by concave increasing functions $\phi\colon (0,\infty)\to(0,\infty)$. Two separated nets are called $\phi$-displacement equivalent if, roughly speaking, there is a bijection between them which, for large radii $R$, displaces points of norm at most $R$ by something of order at most $\phi(R)$. We show that the spectrum of $\phi$-displacement equivalence spans from the established notion of bounded displacement equivalence, which corresponds to bounded $\phi$, to the indiscrete equivalence relation, coresponding to $\phi(R)\in \Omega(R)$, in which all separated nets are equivalent. In between the two ends of this spectrum, the notions of $\phi$-displacement equivalence are shown to be pairwise distinct with respect to the asymptotic classes of $\phi(R)$ for $R\to\infty$. We further undertake a comparison of our notion of $\phi$-displacement equivalence with previously studied relations on separated nets. Particular attention is given to the interaction of the notions of $\phi$-displacement equivalence with that of bilipschitz equivalence., Comment: Final version, to appear in Geometriae Dedicata

Kalu\v{z}a, Kopeck\'a and the author have shown that the best Lipschitz constant for mappings taking a given $n^{d}$-element set in the integer lattice $\mathbb{Z}^{d}$, with $n\in \mathbb{N}$, surjectively to the regular $n$ times $n$ grid $\left\{1,\ldots,n\right\}^{d}$ may be arbitrarily large. However, there remain no known, non-trivial asymptotic bounds, either from above or below, on how this best Lipschitz constant grows with $n$. We approach this problem from a probabilistic point of view. More precisely, we consider the random configuration of $n^{d}$ points inside a given finite lattice and establish almost sure, asymptotic upper bounds of order $\log n$ on the best Lipschitz constant of mappings taking this set surjectively to the regular $n$ times $n$ grid $\left\{1,\ldots,n\right\}^{d}$., Comment: 17 pages, main result reformulated and several minor corrections and improvements. To appear in PAFA

We show that the typical nonexpansive mapping on a small enough subset of a CAT($\kappa$)-space is a contraction in the sense of Rakotch. By typical we mean that the set of nonexpansive mapppings without this property is a $\sigma$-porous set and therefore also of the first Baire category. Moreover, we exhibit metric spaces where strict contractions are not dense in the space of nonexpansive mappings. In some of these cases we show that all continuous self-mappings have a fixed point nevertheless., Comment: 14 pages. Accepted version of the manuscript

We obtain a criterion for an analytic subset of a Euclidean space to contain points of differentiability of a typical Lipschitz function, namely, that it cannot be covered by countably many sets, each of which is closed and purely unrectifiable (has zero length intersection with every $C^1$ curve). Surprisingly, we establish that any set failing this criterion witnesses the opposite extreme of typical behaviour: In any such coverable set a typical Lipschitz function is everywhere severely non-differentiable., Comment: Accepted for publication in Forum of Mathematics Sigma. Some revisions made according to the referee's report

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 $\rho\colon [0,1]^{d}\to(0,\infty)$. In the present work we obtain stronger types of non-realisable densities., Comment: 52 pages. A part of this work is an extensive refinement of a part of arXiv:1704.01940. v4: Changes according to referee's comments, correction of minor typos and errors, especially in the volume bound in Lemma 4.10. To appear in the Israel Journal of Mathematics

We verify the existence of a purely unrectifiable set in which the typical Lipschitz function has a large set of full differentiability points. The example arises from a construction, due to Cs\"ornyei, Preiss and Ti\v{s}er, of a universal differentiability set in which a certain Lipschitz function has only a purely unrectifiable set of differentiability points., Comment: 33 pages, Appendix 6 pages Accepted for publication in Journal of Mathematical Analysis and Applications. Some revisions made following the referees' feedback

We prove that the regular $n\times n$ square grid of points in the integer lattice $\mathbb{Z}^{2}$ cannot be recovered from an arbitrary $n^{2}$-element subset of $\mathbb{Z}^{2}$ 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., Comment: 60 pages (43 pages of the main part, 13 pages of appendices), 10 figures. This is a revised version according to referees' comments. Our version of the proof of the theorem about bilipschitz decomposition of Lipschitz regular mappings was greatly simplified. To appear in GAFA

We prove that universal differentiability sets in Euclidean spaces possess distinctive structural properties. Namely, we show that any universal differentiability set contains a `kernel' in which the points of differentiability of each Lipschitz function are dense. We further prove that no universal differentiability set may be decomposed as a countable union of relatively closed, non-universal differentiability sets. The sharpness of this result, with respect to existing decomposibility results of the opposite nature, is discussed.

We consider a large class of geodesic metric spaces, including Banach spaces, hyperbolic spaces and geodesic $\mathrm{CAT}(\kappa)$-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 $\sigma$-porous subset. For separable 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., Comment: 35 pages; accepted version of the manuscript; accepted for publication in Topological Methods in Nonlinear Analysis

We consider the space of non-expansive mappings on a bounded, closed and convex subset of a Banach space equipped with the metric of uni- form convergence. We show that the set of strict contractions is a {\sigma}-porous subset., Comment: Accepted version

We give a constructive proof that any $\sigma$-porous subset of a Hilbert space has Lebesgue measure zero on typical $C^{1}$ curves. Further, we discover that this result does not extend to all forms of porosity; we find that even power-$p$ porous sets may meet many $C^{1}$ curves in positive measure.

We study notions of absolute continuity for functions defined on $\mathbb{R}^n$similar to the notion of $\alpha$-absolute continuity in the sense of Bongiorno. We confirm a conjecture of Mal\'y that 1-absolutely continuous functions do not need to be differentiable a.e., and we show several other pathological examples of functions in this class. We establish containment relations of the class $1-AC_{\rm WDN}$ which consits of all functions in $1-AC$ which are in the Sobolev space $W^{1,2}_{loc}$, are differentiable a.e. and satisfy the Luzin (N) property, with previously studied classes of absolutely continuous functions., Comment: 29 pages, 2 figures, this is a new version which includes a new co-author and new results, one of the results from the previous version was separated into a different paper, since it is of independent interest and to reduce the length of the paper (which even now is 29 pages long)

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., Comment: 23 pages