A characterization of homothetic simplices (Q1302035)

The author proves the following theorem: For convex bodies \(K_1, K_2\subset E^d,\) \(d\geq 1,\) the following conditions are equivalent: (1) \(K_1\) and \(K_2\) are homothetic \(d\)-simplices; (2) the \(d\)-dimensional intersections \(K_1\cap(z + K_2)\), \(z\in E^d,\) belong to a unique homothety class of convex bodies; (3) the \(d\)-dimensional intersections \(K_1\cap(z+ K_2)\), \(z\in E^d\), belong to at most countably many homothety classes of convex bodies.