When midsets are manifolds
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 Proc. Amer. Math. Soc. 61 (1976), 353360 Request permission
Abstract:
The midset $M$ of two disjoint closed subsets $A$ and $B$ of $n$space ${E^n}$ is defined as the set of all points of ${E^n}$ having equal distances to both $A$ and $B$. Such midsets are not always manifolds, but when either $A$ or $B$ is a convex set it follows that $M$ is homeomorphic to an open subset of an $(n  1)$sphere ${S^{n  1}}$. Furthermore, in this situation $M$ will be homeomorphic to ${S^{n  1}}$ if and only if the convex set $A$ is bounded and lies in the interior of the convex hull $C(B)$ of $B$. If $A$ is a singleton set and $r$ is the dimension of the smallest Euclidean flat $P$ in ${E^n}$ containing $A \cup B$, then $P \cap M$ is an $(r  1)$sphere or an open $(r  1)$cell depending upon whether or not $A$ lies in the interior (relative to $P$) of $C(B)$. In either case $M = (P \cap M) \times {E^{n  r}}$. A manifold lying in a midset in ${E^3}$ is always tamely embedded, as are $\varepsilon$boundaries of certain special subsets of ${E^n}$.References

H. Bell, Some topological extensions of plane geometry (manuscript).
 C. E. Burgess and J. W. Cannon, Embeddings of surfaces in $E^{3}$, Rocky Mountain J. Math. 1 (1971), no.Β 2, 259β344. MR 278277, DOI 10.1216/RMJ197112259
 Herbert Busemann, The geometry of geodesics, Academic Press, Inc., New York, N.Y., 1955. MR 0075623
 Morton Brown, Sets of constant distance from a planar set, Michigan Math. J. 19 (1972), 321β323. MR 315714
 James W. Cannon, $^{\ast }$taming sets for crumpled cubes. I. Basic properties, Trans. Amer. Math. Soc. 161 (1971), 429β440. MR 282353, DOI 10.1090/S00029947197102823537
 A. V. ΔernavskiΔ, The identity of local flatness and local simple connectedness for imbeddings of $(n1)$dimensional into $n$dimensional manifolds when $n>4$, Mat. Sb. (N.S.) 91(133) (1973), 279β286, 288 (Russian). MR 0334222
 Robert J. Daverman, Locally nice codimension one manifolds are locally flat, Bull. Amer. Math. Soc. 79 (1973), 410β413. MR 321095, DOI 10.1090/S000299041973131908 β, Sewings of closed $n$cellcomplements, Trans. Amer. Math. Soc. (to appear).
 Steve Ferry, When $\epsilon$boundaries are manifolds, Fund. Math. 90 (1975/76), no.Β 3, 199β210. MR 413112, DOI 10.4064/fm903199210
 H. C. Griffith, Spheres uniformly wedged between balls are tame in $E^{3}$, Amer. Math. Monthly 75 (1968), 767. MR 234436, DOI 10.2307/2315205
 Ronald Gariepy and W. D. Pepe, On the level sets of a distance function in a Minkowski space, Proc. Amer. Math. Soc. 31 (1972), 255β259. MR 287442, DOI 10.1090/S00029939197202874425
 Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
 Lawrence R. Weill, A new characterization of tame $2$spheres in $E^{3}$, Trans. Amer. Math. Soc. 190 (1974), 243β252. MR 339188, DOI 10.1090/S00029947197403391889
 J. B. Wilker, Equidistant sets and their connectivity properties, Proc. Amer. Math. Soc. 47 (1975), 446β452. MR 355791, DOI 10.1090/S00029939197503557910
Additional Information
 © Copyright 1976 American Mathematical Society
 Journal: Proc. Amer. Math. Soc. 61 (1976), 353360
 MSC: Primary 57A15
 DOI: https://doi.org/10.1090/S00029939197604383422
 MathSciNet review: 0438342