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Density not realizable as the Jacobian determinant of a bilipschitz map

Kaluza V.

Journal of Applied Analysis

2016

Vol. 22, nr 1

37--47

Are every two separated nets in the plane bilipschitz equivalent? In the late 1990s, Burago and Kleiner and, independently, McMullen resolved this beautiful question negatively. Both solutions are based on a construction of a density function that is not realizable as the Jacobian determinant of a bilipschitz map. McMullen's construction is simpler than the Burago–Kleiner one, and we provide a full proof of its nonrealizability, which has not been available in the literature.