When midsets are manifolds
Author:
L. D. Loveland
Journal:
Proc. Amer. Math. Soc. 61 (1976), 353360
MSC:
Primary 57A15
DOI:
https://doi.org/10.1090/S00029939197604383422
MathSciNet review:
0438342
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Abstract  References  Similar Articles  Additional Information
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 ${E^n}$.

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Additional Information
Keywords:
Bisector,
<!โ MATH $\varepsilon$ โ> <IMG WIDTH="15" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\varepsilon$">boundary,
equidistant set,
Euclidean <IMG WIDTH="18" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img45.gif" ALT="$n$">space,
manifold midset,
midsets,
tame manifolds,
tame surfaces
Article copyright:
© Copyright 1976
American Mathematical Society