# 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} }

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

- Mathematics
- 1994

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

- Mathematics
- 1993

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

- Mathematics
- 1997

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

- Mathematics
- 2006

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

- Mathematics

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

- Education
- 1997

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…