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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



When midsets are manifolds

Author: L. D. Loveland
Journal: Proc. Amer. Math. Soc. 61 (1976), 353-360
MSC: Primary 57A15
MathSciNet review: 0438342
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Abstract: The midset $ M$ of two disjoint closed subsets $ A$ and $ B$ of $ n$-space $ {E^n}$ is defined as the set of all points of $ {E^n}$ having equal distances to both $ A$ and $ B$. Such midsets are not always manifolds, but when either $ A$ or $ B$ is a convex set it follows that $ M$ is homeomorphic to an open subset of an $ (n - 1)$-sphere $ {S^{n - 1}}$. Furthermore, in this situation $ M$ will be homeomorphic to $ {S^{n - 1}}$ if and only if the convex set $ A$ is bounded and lies in the interior of the convex hull $ C(B)$ of $ B$. If $ A$ is a singleton set and $ r$ is the dimension of the smallest Euclidean flat $ P$ in $ {E^n}$ containing $ A \cup B$, then $ P \cap M$ is an $ (r - 1)$-sphere or an open $ (r - 1)$-cell depending upon whether or not $ A$ lies in the interior (relative to $ P$) of $ C(B)$. In either case $ M = (P \cap M) \times {E^{n - r}}$. A manifold lying in a midset in $ {E^3}$ is always tamely embedded, as are $ \varepsilon $-boundaries of certain special subsets of $ {E^n}$.

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Keywords: Bisector, $ \varepsilon $-boundary, equidistant set, Euclidean $ n$-space, manifold midset, midsets, tame manifolds, tame surfaces
Article copyright: © Copyright 1976 American Mathematical Society