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 of two disjoint closed subsets and of -space is defined as the set of all points of having equal distances to both and . Such midsets are not always manifolds, but when either or is a convex set it follows that is homeomorphic to an open subset of an -sphere . Furthermore, in this situation will be homeomorphic to if and only if the convex set is bounded and lies in the interior of the convex hull of . If is a singleton set and is the dimension of the smallest Euclidean flat in containing , then is an -sphere or an open -cell depending upon whether or not lies in the interior (relative to ) of . In either case . A manifold lying in a midset in is always tamely embedded, as are -boundaries of certain special subsets of .

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DOI:
https://doi.org/10.1090/S0002-9939-1976-0438342-2

Keywords:
Bisector,
-boundary,
equidistant set,
Euclidean -space,
manifold midset,
midsets,
tame manifolds,
tame surfaces

Article copyright:
© Copyright 1976
American Mathematical Society