Subsets of \(\mathbb{R}^ n\) with convex midsets (Q1341410)

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.