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What is a disk?

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This paper should be considered as a companion report to F.W. Gehring's survey lectures "Characterizations of quasidisks" given at this Summer School [7]. Notation, definitions and background results are given in that paper. In particular, D is a simply connected proper subdomain of $R^2$ unless otherwise stated and D* denotes the exterior of D in $\overline{R}^2$. Many of the characterizations of quasidisks have been motivated by looking at properties of euclidean disks. It is therefore natural to go back and ask if any of the original properties in fact characterize euclidean disks. We follow the procedure in Gehring's lectures and look at four different categories of properties: 1. Geometric properties, 2. Conformal invariants, 3. Injectivity criteria, 4. Extension properties. As we shall see, the answers are not equally easy to obtain and not always positive. There are, in fact, still many interesting open questions.
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  • Norwegian University of Science and Technology, N-7491 Trondheim, Norway
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