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Given three polyhedra $P$, $Q$, and $R$ in dimensions $a$, $b$, and $c$ respectively, with $a\leq b\leq c$, with the additional condition that: $P=\pi_1(Q)=\pi_2(R)$, where $\pi_1$ and $\pi_2$ are projection maps. Then

Main Question: When is it true that there is a projection map $\pi$ with $Q=\pi(R)$?

(1) What if we consider only (bounded) polytopes?

(2) What if we further restrict to just convex cases?

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    $\begingroup$ Roll-back if my clarifications were too heavy-handed. $\endgroup$
    – Joseph O'Rourke
    Commented Apr 27, 2019 at 23:56

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