Density not realizable as the Jacobian determinant of a bilipschitz map
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Vojtěch Kaluža
Abstract
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.
Funding source: Czech Science Foundation
Award Identifier / Grant number: CE-ITI (P202/12/G061)
Funding source: Ministry of Education, Youth and Sports of the Czech Republic
Award Identifier / Grant number: SVV-2015-260223
Funding statement: The author was supported by the grant CE-ITI (P202/12/G061) of the Czech Science Foundation and by the grant SVV-2015-260223 of the Ministry of Education, Youth and Sports of the Czech Republic.
A preliminary version appeared in Czech as a part of the author's bachelor thesis [5] at the Charles University in Prague in 2012. I would like to thank my supervisor Professor Jiří Matoušek for valuable advice and help he has given me during my studies and in writing this article. I would also like to thank the referees for many helpful comments.
References
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© 2016 by De Gruyter
Artikel in diesem Heft
- Frontmatter
- Admissible pair of spaces for non-correctly solvable linear differential equations
- Defective functions of meromorphic functions in the unit disc
- A boundary-value problem for the equation of mixed type with generalized operators of fractional differentiation in the boundary conditions
- Density not realizable as the Jacobian determinant of a bilipschitz map
- An inequality involving the gamma and digamma functions
- Asymptotic behaviour of solutions to one-dimensional reaction diffusion cooperative systems involving infinitesimal generators
- Generalized slow growth of special monogenic functions
- Iterative algorithm for the split equality problem in Hilbert spaces
- On norm of single layer potentials on segments
Artikel in diesem Heft
- Frontmatter
- Admissible pair of spaces for non-correctly solvable linear differential equations
- Defective functions of meromorphic functions in the unit disc
- A boundary-value problem for the equation of mixed type with generalized operators of fractional differentiation in the boundary conditions
- Density not realizable as the Jacobian determinant of a bilipschitz map
- An inequality involving the gamma and digamma functions
- Asymptotic behaviour of solutions to one-dimensional reaction diffusion cooperative systems involving infinitesimal generators
- Generalized slow growth of special monogenic functions
- Iterative algorithm for the split equality problem in Hilbert spaces
- On norm of single layer potentials on segments
