Convex surfaces which intersect each congruent copy of themselves in a connected set
A convex body \(K\subset {\mathbb{R}}^ 3\) has the intersection property, if for every isometry \(\rho\) of \({\mathbb{R}}^ 3\), the intersection of the boundaries of K and \(\rho\) K is connected. A set \(Y\subset {\mathbb{R}}^ 3\) is the parallel body of a compact convex set X, if there is a Euclidean ball B with the origin as centre, such that \(Y=X+B\). The author presents the following pretty result: If a convex body \(K\subset {\mathbb{R}}^ 3\) has the intersection property, then it is either a Euclidean ball or a parallel body of a line segment.
scientific article; zbMATH DE number 3969529