# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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by L. D. Loveland
Proc. Amer. Math. Soc. 61 (1976), 353-360 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 ${E^n}$.
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