Projections and generated cones of homothetic convex sets

We prove that polyhedra P-1 and P-2 in R-n , n >= 3, are homothetic if and only if their orthogonal projections on every 2-dimensional plane of R-n are homothetic (this assertion does not hold for arbitrary closed convex sets). Also, we show that closed convex sets K-1 and K-2 in R-n are directly homothetic if and only if for any point p(1) is an element of R-n\K-1 there is a point p2 is an element of R-n\K-2 such that the generated cone C-p1(K-1) is a translate of the respective cone C-p2(K-2).