Separated nets in Euclidean space and Jacobians of biLipschitz maps. (English) Zbl 0902.26004

Burago, D.; Kleiner, B.

Geom. Funct. Anal. 8, No. 2, 273-282 (1998).

A subset \(X\) of a metric space \(Z\) is a separated net if there are constants \(a,b> 0\) such that \(d(x,x')> a\) for every pair \(x,x'\in X\) and \(d(z,X)< b\) for every \(z\in Z\).
The authors show the existence of separated nets in the Euclidean plane which are not bi-Lipschitz equivalent to the integer lattice. The argument is based on the construction of a continuous function which is not the Jacobian of a bi-Lipschitz map.

Reviewer: I.S.Molchanov (Glasgow)

MSC:

26B10

Implicit function theorems, Jacobians, transformations with several variables

Keywords:

separated net; integer lattice; continuous function; Jacobian; bi-Lipschitz map