Annales Academię Scientiarum Fennicę

Mathematica

Volumen 33, 2008, 205-230

# EUCLIDEAN QUASICONVEXITY

## Hrant Hakobyan and David A. Herron

Stony Brook University, Department of Mathematics

Stony Brook, NY 11794-3651, U.S.A.; hhakob 'at' math.sunysb.edu

University of Cincinnati, Department of Mathematical Sciences

839 Old Chemistry Building,
Cincinnati, OH 45221-0025, U.S.A.;
David.Herron 'at' math.UC.edu

**Abstract.**
We exhibit compact totally disconnected sets in **R**^{n},
with Hausdorff dimension *n* - 1, whose complements fail to be
quasiconvex, and similar sets with positive *n*-measure whose
complements are quasiconvex. We characterize the
finitely connected quasiconvex plane domains.
We present related results for bounded turning.

**2000 Mathematics Subject Classification:**
Primary 30C655; Secondary 51F99.

**Key words:**
Quasiconvex, Jordan curve,
Hausdorff dimension.

**Reference to this article:** H. Hakobyan and D.A. Herron:
Euclidean quasiconvexity.
Ann. Acad. Sci. Fenn. Math. 33 (2008), 205-230.

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