When Midsets are Manifolds
L. D. Loveland · Proceedings of the American Mathematical Society · 1976
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 ${E^n}$.