# Mapping $n$ grid points onto a square forces an arbitrarily large Lipschitz constant

## TL;DR

This paper proves that mapping an $n$-grid onto a square grid with a fixed Lipschitz constant is impossible for large $n$, answering a question from 2002 and advancing understanding of Lipschitz mappings and non-realisable densities.

## Contribution

It provides a negative answer to Feige's 2002 question by analyzing Lipschitz mappings in a continuous setting and exploring properties of non-realisable densities in function spaces.

## Key findings

- Regular $n imes n$ grid cannot be recovered via Lipschitz maps with fixed constant
- Established new results on bilipschitz decomposability of Lipschitz maps
- Showed prevalence of strongly non-realisable densities in function spaces

## 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.