On polytopes with congruent projections or sections

Let $2\le k\le d-1$ and let $P$ and $Q$ be two convex polytopes in ${\mathbb E^d}$. 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, $P \cap H$, $Q \cap H$, by every $k$-dimensional subspace $H$, are congruent.