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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.

##### MSC:
 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces) 54E45 Compact (locally compact) metric spaces
##### Keywords:
sphere; $$n$$-cell; ball; midset; convex body
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##### References:
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