On polytopes with congruent projections or sections

Let 2≤k≤d−1 and let P and Q be two convex polytopes in Ed. Assume that their projections, P|H, Q|H, onto every k-dimensional subspace H, are congruent. In this paper we show that P and Q or P and −Q are translates of each other. We also prove an analogous result for sections by showing that P=Q or P=−Q, provided the polytopes contain the origin in their interior and their sections, P∩H, Q∩H, by every k-dimensional subspace H, are congruent.