Description of a class of solids in $\mathbb R^3$

Let $Q\subset\mathbb R^3$ be a compact convex solid such that for each parallel (not necessarily orthogonal) projection onto any plane no two antipodal faces of the solid are projected strictly inside the projection of all $Q$. Then $Q$ is either a cone with a convex base or a frustrum of a trihedral pyramid or a prism (possibly with nonparallel bases).