Mapping n Grid Points Onto a Square Forces an Arbitrarily Large Lipschitz Constant

Abstract

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.

Highly irregular separated nets

“…We will also need two auxiliary lemmas on weak convergence of measures which are probably a common part of knowledge in measure theory. Their proofs can be found in [5].…”

“…The statement is an amalgamation of Lemmas 4.1 and 4.2. Constructions of non-realisable densities based on statements of this type have already been written in great detail in [5] and originally in [2]. Therefore, following Lemma 3.5 we only give an informal sketch of the proof of Theorem 1.2.…”

“…In this sense, almost all continuous functions are not Lipschitz regular realisable. In the same paper [5], they also showed that the set of L ∞ densities that are bilipschitz realisable in the sense of (1.2) is σ-porous. Independently, results in this direction were also obtained by Viera [12].…”

“…In fact, Burago and Kleiner [2] produced ρ that is, in addition, continuous. In [5] the authors and Kopecká strengthened these results and established that the set of those continuous functions inside the space C(I d ) with the supremum norm that admit a Lipschitz regular solution f :…”

Divergence of separated nets with respect to displacement equivalence

“…Proof. If property (i) is omitted, the proof is contained in [1, Proof of Lemma 2.1]; similar constructions are also given in [3] and [2]. Getting property (i) only requires taking a little extra care in the construction of [1, Proof of Lemma 2.1].…”