## Sets of constant distance from a compact set in 2-manifolds with a geodesic metric

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- by Alexander Blokh, Michał Misiurewicz and Lex Oversteegen
- Proc. Amer. Math. Soc.
**137**(2009), 733-743 - DOI: https://doi.org/10.1090/S0002-9939-08-09502-6
- Published electronically: October 8, 2008
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## Abstract:

Let $(M, d)$ be a complete topological 2-manifold, possibly with boundary, with a geodesic metric $d$. Let $X\subset M$ be a compact set. We show then that for all but countably many $\varepsilon$ each component of the set $S(X, \varepsilon )$ of points $\varepsilon$-distant from $X$ is either a point, a simple closed curve disjoint from $\partial M$ or an arc $A$ such that $A\cap \partial M$ consists of both endpoints of $A$ and that arcs and simple closed curves are dense in $S(X, \varepsilon )$. In particular, if the boundary $\partial M$ of $M$ is empty, then each component of the set $S(X, \varepsilon )$ is either a point or a simple closed curve and the simple closed curves are dense in $S(X, \varepsilon )$.## References

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## Bibliographic Information

**Alexander Blokh**- Affiliation: Department of Mathematics, University of Alabama in Birmingham, University Station, Birmingham, Alabama 35294-2060
- MR Author ID: 196866
- Email: ablokh@math.uab.edu
**Michał Misiurewicz**- Affiliation: Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216
- MR Author ID: 125475
- Email: mmisiure@math.iupui.edu
**Lex Oversteegen**- Affiliation: Department of Mathematics, University of Alabama in Birmingham, University Station, Birmingham, Alabama 35294-2060
- MR Author ID: 134850
- Email: overstee@math.uab.edu
- Received by editor(s): February 8, 2007
- Received by editor(s) in revised form: January 3, 2008
- Published electronically: October 8, 2008
- Additional Notes: The first author was partially supported by NSF grant DMS 0456748

The second author was partially supported by NSF grant DMS 0456526

The third author was partially supported by NSF grant DMS 0405774 - Communicated by: Alexander N. Dranishnikov
- © Copyright 2008 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**137**(2009), 733-743 - MSC (2000): Primary 54E35, 54F15
- DOI: https://doi.org/10.1090/S0002-9939-08-09502-6
- MathSciNet review: 2448596