zbMATH — the first resource for mathematics

Subsets of \(\mathbb{R}^ n\) with convex midsets. (English) Zbl 0817.52006
The set of all points of a subset \(X\) of Euclidean \(n\)-space \(E^ n\) which are equidistant from distinct points \(x\) and \(y\) of \(X\) is denoted by \(M(x,y)\) and it is called a midset. By a nondegenerate set the authors mean a set containing more than one point. A theorem says that if for every two distinct points \(x\) and \(y\) of a nondegenerate subset \(X\) of \(E^ n\), where \(n \geq 2\), the midset \(M(x,y)\) is a convex \((n - 1)\)- cell, then \(X\) is a convex \(n\)-cell. (The authors do not define the notion of the convex \(k\)-cell; from the context it follows that it is a subset of \(E^ n\) isometric to a convex body in \(E^ k\), wher \(k \leq n\).) Another theorem says that if \(X\) is a nondegenerate compact subset of \(E^ n\), where \(n \geq 3\), and if for every pair of distinct points \(x\), \(y \in X\) the midset \(M(x,y)\) is the boundary of a convex \((n - 1)\)- cell, then \(X\) is the boundary of a convex \(n\)-cell.

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
54E45 Compact (locally compact) metric spaces
PDF BibTeX Cite
Full Text: DOI
[1] Berard, A. D.; Nitka, W., A new definition of the circle by use of bisectors, Fund. Math., 85, 49-55, (1974) · Zbl 0281.53042
[2] Loveland, L. D., Metric spaces with connected midsets, Houston J. Math., 3, 495-501, (1977) · Zbl 0375.54020
[3] Loveland, L. D., A metric characterization of a simple closed curve, Gen. Topology Appl., 6, 309-313, (1976) · Zbl 0323.54030
[4] Loveland, L. D., The double midset conjecture for continua in the plane, Gen. Topology Appl., 40, 117-129, (1991) · Zbl 0735.54020
[5] Loveland, L. D.; Valentine, J. E., A characterization of 2-dimensional spherical space, Proc. Amer. Math. Soc., 38, 598-604, (1973) · Zbl 0257.52017
[6] Loveland, L. D.; Valentine, J. E., Characterizing a circle with the double midset property, Proc. Amer. Math. Soc., 53, 443-444, (1975) · Zbl 0317.52003
[7] Loveland, L. D.; Wayment, S. G., Characterizing a curve with the double midset property, Amer. Math. Monthly, 81, 1003-1006, (1974) · Zbl 0291.54042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.