On polytopes with congruent projections or sections

Let $2\le k\le d-1$ and let $P$ and $Q$ be two convex polytopes in ${\mathbb E^d}$\mathrm{.}Assumethattheirprojections,$P|H$,$Q|H$,ontoevery$k$-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.