When midsets are manifolds

Author:
L. D. Loveland

Journal:
Proc. Amer. Math. Soc. **61** (1976), 353-360

MSC:
Primary 57A15

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

MathSciNet review:
0438342

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 .

**[1]**H. Bell,*Some topological extensions of plane geometry*(manuscript).**[2]**C. E. Burgess and J. W. Cannon,*Embeddings of surfaces in*, Rocky Mountain J. Math.**1**(1971), 259-344. MR**43**#4008. MR**0278277 (43:4008)****[3]**H. Busemann,*The geometry of geodesics*, Academic Press, New York, 1955. MR**17**, 779. MR**0075623 (17:779a)****[4]**Morton Brown,*Sets of constant distance from a planar set*, Michigan Math. J.**19**(1972), 321-323. MR**47**#4263. MR**0315714 (47:4263)****[5]**J. W. Cannon, -*taming sets for crumpled cubes*. I.*Basic properties*, Trans. Amer. Math. Soc.**161**(1971), 429-440. MR**43**#8065. MR**0282353 (43:8065)****[6]**A. V. Černavskiĭ,*Coincidence of local flatness and local simple-connectedness for embeddings of*-*dimensional manifolds in*-*dimensional manifolds when*, Mat. Sb. (N.S.)**91**(**133**) (1973), 279-286, 288 = Math. USSR Sbornik**20**(1973), 297-304. MR**48**#12541. MR**0334222 (48:12541)****[7]**R. J. Daverman,*Locally nice codimension one manifolds are locally flat*, Bull. Amer. Math. Soc.**79**(1973), 410-413. MR**47**#9628. MR**0321095 (47:9628)****[8]**-,*Sewings of closed*-*cell-complements*, Trans. Amer. Math. Soc. (to appear).**[9]**Steve Ferry,*When*-*boundaries are manifolds*, Fund. Math. (to appear). MR**0413112 (54:1233)****[10]**H. C. Griffith,*Spheres uniformly wedged between balls are tame in*, Amer. Math. Monthly**75**(1968), 767. MR**38**#2753. MR**0234436 (38:2753)****[11]**R. Gariepy and W. D. Pepe,*On the level sets of a distance function in a Minkowski space*, Proc. Amer. Math. Soc.**31**(1972), 255-259. MR**44**#4646. MR**0287442 (44:4646)****[12]**W. Hurewicz and H. Wallman,*Dimension theory*, Princeton Univ. Press, Princeton, N.J., 1941. MR**3**, 312. MR**0006493 (3:312b)****[13]**L. R. Weill,*A new characterization of tame*-*spheres in*, Trans. Amer. Math. Soc.**190**(1974), 243-252. MR**49**#3951. MR**0339188 (49:3951)****[14]**J. B. Wilker,*Equidistant sets and their connectivity properties*, Proc. Amer. Math. Soc.**47**(1975), 446-452. MR**50**#8265. MR**0355791 (50:8265)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
57A15

Retrieve articles in all journals with MSC: 57A15

Additional Information

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