When midsets are manifolds
Author:
L. D. Loveland
Journal:
Proc. Amer. Math. Soc. 61 (1976), 353360
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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 Morton Brown, Sets of constant distance from a planar set, Michigan Math. J. 19 (1972), 321323. MR 47 #4263. MR 0315714 (47:4263)
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 J. W. Cannon, taming sets for crumpled cubes. I. Basic properties, Trans. Amer. Math. Soc. 161 (1971), 429440. MR 43 #8065. MR 0282353 (43:8065)
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 A. V. Černavskiĭ, Coincidence of local flatness and local simpleconnectedness for embeddings of dimensional manifolds in dimensional manifolds when , Mat. Sb. (N.S.) 91 (133) (1973), 279286, 288 = Math. USSR Sbornik 20 (1973), 297304. MR 48 #12541. MR 0334222 (48:12541)
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 , Sewings of closed cellcomplements, Trans. Amer. Math. Soc. (to appear).
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 H. C. Griffith, Spheres uniformly wedged between balls are tame in , Amer. Math. Monthly 75 (1968), 767. MR 38 #2753. MR 0234436 (38:2753)
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 R. Gariepy and W. D. Pepe, On the level sets of a distance function in a Minkowski space, Proc. Amer. Math. Soc. 31 (1972), 255259. MR 44 #4646. MR 0287442 (44:4646)
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 W. Hurewicz and H. Wallman, Dimension theory, Princeton Univ. Press, Princeton, N.J., 1941. MR 3, 312. MR 0006493 (3:312b)
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 L. R. Weill, A new characterization of tame spheres in , Trans. Amer. Math. Soc. 190 (1974), 243252. MR 49 #3951. MR 0339188 (49:3951)
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 J. B. Wilker, Equidistant sets and their connectivity properties, Proc. Amer. Math. Soc. 47 (1975), 446452. MR 50 #8265. MR 0355791 (50:8265)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197604383422
PII:
S 00029939(1976)04383422
Keywords:
Bisector,
boundary,
equidistant set,
Euclidean space,
manifold midset,
midsets,
tame manifolds,
tame surfaces
Article copyright:
© Copyright 1976
American Mathematical Society
