A characterization of translated convex bodies

In this work we present a theorem regarding two convex bodies $K_1,K_2\subset {\mathbb{R}}^n$, $n\ge 3$, and two families of sections of them, given by two families of tangent planes of two spheres $S_i\subset \text{int }{K}_i$, $i=1,2$ such that, for every pair ${\mathrm{\Pi }}_1$, ${\mathrm{\Pi }}_2$ of parallel supporting planes of $S_1$, $S_2$, respectively, which are corresponding (this means, that the outer normal vectors of the supporting half spaces determined by the two planes have the same direction), the sections ${\mathrm{\Pi }}_1\cap K_1$, ${\mathrm{\Pi }}_2\cap K_2$ are translated, the theorem claims that if $S_1$, $S_2$ have the same radius, the bodies are translated, otherwise, the bodies are also spheres.