When midsets are manifolds
Author:
L. D. Loveland
Journal:
Proc. Amer. Math. Soc. 61 (1976), 353360
MSC:
Primary 57A15
MathSciNet review:
0438342
Fulltext 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
(43 #4008)
 [3]
Herbert
Busemann, The geometry of geodesics, Academic Press Inc., New
York, N. Y., 1955. MR 0075623
(17,779a)
 [4]
Morton
Brown, Sets of constant distance from a planar set, Michigan
Math. J. 19 (1972), 321–323. MR 0315714
(47 #4263)
 [5]
James
W. Cannon, *taming sets for crumpled cubes. I.
Basic properties, Trans. Amer. Math. Soc.
161 (1971),
429–440. MR 0282353
(43 #8065), http://dx.doi.org/10.1090/S00029947197102823537
 [6]
A.
V. Černavskiĭ, The identity of local flatness and
local simple connectedness for imbeddings of (𝑛1)dimensional into
𝑛dimensional manifolds when 𝑛>4, Mat. Sb. (N.S.)
91(133) (1973), 279–286, 288 (Russian). MR 0334222
(48 #12541)
 [7]
Robert
J. Daverman, Locally nice codimension one manifolds
are locally flat, Bull. Amer. Math. Soc. 79 (1973), 410–413.
MR
0321095 (47 #9628), http://dx.doi.org/10.1090/S000299041973131908
 [8]
, Sewings of closed cellcomplements, Trans. Amer. Math. Soc. (to appear).
 [9]
Steve
Ferry, When 𝜀boundaries are manifolds, Fund. Math.
90 (1975/76), no. 3, 199–210. MR 0413112
(54 #1233)
 [10]
H.
C. Griffith, Spheres uniformly wedged between balls are tame in
𝐸³, Amer. Math. Monthly 75 (1968), 767.
MR
0234436 (38 #2753)
 [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
(44 #4646), http://dx.doi.org/10.1090/S00029939197202874425
 [12]
Witold
Hurewicz and Henry
Wallman, Dimension Theory, Princeton Mathematical Series, v.
4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
(3,312b)
 [13]
Lawrence
R. Weill, A new characterization of tame
2spheres in 𝐸³, Trans. Amer.
Math. Soc. 190
(1974), 243–252. MR 0339188
(49 #3951), http://dx.doi.org/10.1090/S00029947197403391889
 [14]
J.
B. Wilker, Equidistant sets and their
connectivity properties, Proc. Amer. Math.
Soc. 47 (1975),
446–452. MR 0355791
(50 #8265), http://dx.doi.org/10.1090/S00029939197503557910
 [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), 259344. 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), 321323. 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), 429440. MR 43 #8065. MR 0282353 (43:8065)
 [6]
 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)
 [7]
 R. J. Daverman, Locally nice codimension one manifolds are locally flat, Bull. Amer. Math. Soc. 79 (1973), 410413. MR 47 #9628. MR 0321095 (47:9628)
 [8]
 , Sewings of closed cellcomplements, 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), 255259. 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), 243252. MR 49 #3951. MR 0339188 (49:3951)
 [14]
 J. B. Wilker, Equidistant sets and their connectivity properties, Proc. Amer. Math. Soc. 47 (1975), 446452. MR 50 #8265. MR 0355791 (50:8265)
Similar Articles
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/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
