[math-fun] Five geometry problems

Let V be the set of vertices of one of the Platonic solids in 3-space. QUESTION: For which 3D rotations of V does the orthogonal projection P of V to the xy-plane realize the maximum possible minimum distance among points of P(V) ? E.g., for the tetrahedron this might occur only when the vertices project to the corners of a square. (Only essentially distinct solutions are of interest: rotations for which the projections of V are not isometric.) For definiteness we could assume that V lies on the unit sphere. --Dan P.S. For a warmup, the same question could be asked about orthogonal projections to the x-axis. Also, a related question is to minimize the maximum possible distance: find the projection of V with the smallest diameter.