Bi-Lipschitz Decomposition of Lipschitz functions into a Metric space

@article{Schul2007BiLipschitzDO,
  title={Bi-Lipschitz Decomposition of Lipschitz functions into a Metric space},
  author={Raanan Schul},
  journal={Revista Matematica Iberoamericana},
  year={2007},
  volume={25},
  pages={521-531}
}
  • R. Schul
  • Published 22 February 2007
  • Mathematics
  • Revista Matematica Iberoamericana
We prove a quantitative version of the following statement. Given a Lipschitz function f from the k-dimensional unit cube into a general metric space, one can decomposed f into a finite number of BiLipschitz functions f|_{F_i} so that the k-Hausdorff content of f([0,1]^k\setminus \cup F_i) is small. We thus generalize a theorem of P. Jones (1988) from the setting of R^d to the setting of a general metric space. This positively answers problem 11.13 in ``Fractured Fractals and Broken Dreams" by… 
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References

SHOWING 1-10 OF 14 REFERENCES
Rectifiable metric spaces: local structure and regularity of the Hausdorff measure
We consider the question whether the "nice" density behaviour of Hausdorff measure on rectifiable subsets of Euclidian spaces preserves also in the general metric case. For this purpose we show the
Analysis of and on uniformly rectifiable sets
The notion of uniform rectifiability of sets (in a Euclidean space), which emerged only recently, can be viewed in several different ways. It can be viewed as a quantitative and scale-invariant
Fractured fractals and broken dreams : self-similar geometry through metric and measure
1. Basic definitions 2. Examples 3. Comparison 4. The Heisenberg group 5. Background information 6. Stronger self-similarity for BPI spaces 7. BPI equivalence 8. Convergence of metric spaces 9. Weak
Ahlfors-Regular Curves In Metric Spaces
We discuss 1-Ahlfors-regular connected sets in a general metric space and prove that such sets are `flat' on most scales and in most locations. Our result is quantitative, and when combined with work
Analyst ’ s Traveling Salesman Theorems . A Survey
The purpose of this essay is to present a partial survey of a family of theorems that are usually referred to as analyst’s traveling salesman theorems (also referred to as geometric traveling
Thirty-three yes or no questions about mappings, measures, and metrics
Most problems in the ensuing list are of fairly recent origin. None of them seem easy and some are likely to be very difficult. The formulation of each problem is such that it can be answered by one
Lipschitz and bi-Lipschitz Functions.
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1
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