Title: Mapping $n$ grid points onto a square forces an arbitrarily large Lipschitz constant,
Abstract: 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.