When midsets are manifolds

Author:
L. D. Loveland

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

MSC:
Primary 57A15

MathSciNet review:
0438342

Full-text PDF Free Access

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 𝐸³*, Rocky Mountain J. Math.**1**(1971), no. 2, 259–344. MR**0278277****[3]**Herbert Busemann,*The geometry of geodesics*, Academic Press Inc., New York, N. Y., 1955. MR**0075623****[4]**Morton Brown,*Sets of constant distance from a planar set*, Michigan Math. J.**19**(1972), 321–323. MR**0315714****[5]**James W. Cannon,**-taming sets for crumpled cubes. I. Basic properties*, Trans. Amer. Math. Soc.**161**(1971), 429–440. MR**0282353**, 10.1090/S0002-9947-1971-0282353-7**[6]**A. V. Černavskiĭ,*The identity of local flatness and local simple connectedness for imbeddings of (𝑛-1)-dimensional into 𝑛-dimensional manifolds when 𝑛>4*, Mat. Sb. (N.S.)**91(133)**(1973), 279–286, 288 (Russian). MR**0334222****[7]**Robert J. Daverman,*Locally nice codimension one manifolds are locally flat*, Bull. Amer. Math. Soc.**79**(1973), 410–413. MR**0321095**, 10.1090/S0002-9904-1973-13190-8**[8]**-,*Sewings of closed*-*cell-complements*, Trans. Amer. Math. Soc. (to appear).**[9]**Steve Ferry,*When 𝜀-boundaries are manifolds*, Fund. Math.**90**(1975/76), no. 3, 199–210. MR**0413112****[10]**H. C. Griffith,*Spheres uniformly wedged between balls are tame in 𝐸³*, Amer. Math. Monthly**75**(1968), 767. MR**0234436****[11]**Ronald 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**0287442**, 10.1090/S0002-9939-1972-0287442-5**[12]**Witold Hurewicz and Henry Wallman,*Dimension Theory*, Princeton Mathematical Series, v. 4, Princeton University Press, Princeton, N. J., 1941. MR**0006493****[13]**Lawrence R. Weill,*A new characterization of tame 2-spheres in 𝐸³*, Trans. Amer. Math. Soc.**190**(1974), 243–252. MR**0339188**, 10.1090/S0002-9947-1974-0339188-9**[14]**J. B. Wilker,*Equidistant sets and their connectivity properties*, Proc. Amer. Math. Soc.**47**(1975), 446–452. MR**0355791**, 10.1090/S0002-9939-1975-0355791-0

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

Retrieve articles in all journals with MSC: 57A15

Additional Information

DOI:
http://dx.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