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A central set of dimension $ 2$


Authors: Christopher J. Bishop and Hrant Hakobyan
Journal: Proc. Amer. Math. Soc. 136 (2008), 2453-2461
MSC (2000): Primary 28A78; Secondary 28A75
DOI: https://doi.org/10.1090/S0002-9939-08-09173-9
Published electronically: March 7, 2008
MathSciNet review: 2390513
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Abstract: The central set of a domain $ D$ is the set of centers of maximal discs in $ D$. Fremlin proved that the central set of a planar domain has zero area and asked whether it can have Hausdorff dimension strictly larger than $ 1$. We construct a planar domain with central set of Hausdorff dimension $ 2$.


References [Enhancements On Off] (What's this?)

  • 1. C. J. Bishop, Conformal mapping in linear time, preprint, 2006.
  • 2. L.A. Caffarelli, A. Friedman, The free boundary for elastic-plastic torsion problems, Trans. Amer. Math. Soc. 252 (1979), 65-97. MR 534111 (80i:35059)
  • 3. H. I. Choi, S. W. Choi and H. P. Moon. Mathematical theory of medial axis transform, Pacific J. Math. (1) 181 (1997), 57-88. MR 1491036 (99m:53008)
  • 4. P. Erdős, Some remarks on measurability of certain sets, Bull. Amer. Math. Soc. 51 (1945), 728-731. MR 0013776 (7:197f)
  • 5. W. D. Evans, D. J. Harris, Sobolev embeddings for generalized ridged domains, Proc. London Math. Soc. (3) 54 (1987), 141-175. MR 872254 (88b:46056)
  • 6. D. H. Fremlin, Skeletons and central sets, Proc. London Math. Soc. (3) 74 (1997), no. 3, 701-720. MR 1434446 (97m:54059)
  • 7. P. Mattila, Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. MR 1333890 (96h:28006)
  • 8. T. W. Ting, The ridge of a Jordan domain and completely plastic torsion, J. Math. Mech. 15 (1966), 15-47. MR 0184503 (32:1975)

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Additional Information

Christopher J. Bishop
Affiliation: Department of Mathematics, SUNY Stony Brook, Stony Brook, New York 11790
Email: bishop@math.sunysb.edu

Hrant Hakobyan
Affiliation: Department of Mathematics, University of Toronto, Toronto, ON, Canada M5S 2E4

DOI: https://doi.org/10.1090/S0002-9939-08-09173-9
Keywords: Hausdorff dimension, central set, medial axis, skeleton, Lipschitz domain, maximal discs, disc trees
Received by editor(s): February 1, 2007
Published electronically: March 7, 2008
Additional Notes: The first author was partially supported by NSF Grant DMS 04-05578.
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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