**Conventions:** A *polytope* in a finite-dimensional $\mathbb R$-vector space $V$ is defined to be a convex hull of finitely many points in $V$. A *polyhedron* in a finite-dimensional $\mathbb R$-vector space $V$ is defined to be an intersection of finitely many closed halfspaces in $V$ (that is, the set of solutions of a system of finitely many non-strict linear inequalities on a vector in $V$). It is known that the polytopes are exactly the bounded polyhedra.

Note that, for me, $\mathbb R$ really can mean any ordered field, like $\mathbb Q$ or $\mathbb Q\left[\sqrt{2}\right]$ or many others. Every claim stated below holds when $\mathbb R$ is replaced by any ordered field, and an answer that makes use of special properties of $\mathbb R$ is welcomed but won't be considered final.

**Background:** The decomposition theorem for polyhedra yields the following facts as easy consequences:

**1.** If $f:V\to W$ is an $\mathbb R$-linear map between finite-dimensional $\mathbb R$-vector spaces, and $P$ is a polyhedron in $V$, then $f\left(P\right)$ is a polyhedron. (The same statement holds with "polyhedron" replaced by "polytope", but that is a triviality.)

**2.** If $f:V\to W$ is an $\mathbb R$-linear map between finite-dimensional $\mathbb R$-vector spaces, and $P$ is a polyhedron in $W$, then $f^{-1}\left(P\right)$ is a polyhedron. (This one is obvious, but just mentioned here for the sake of "symmetry".)

**3.** If $P$ is a polyhedron in a finite-dimensional $\mathbb R$-vector space $V$, and $Q$ is a polyhedron in a finite-dimensional $\mathbb R$-vector space $W$, then $P\times Q$ is a polyhedron in $V\times W$. (The same holds for polytopes.)

**4.** If $P$ and $Q$ are two polyhedra in one and the same finite-dimensional $\mathbb R$-vector space, then the Minkowski sum $P+Q$ and the intersection $P\cap Q$ are polyhedra as well. (Again, the same holds for polytopes.)

**5.** If $P$ is a polyhedron in a finite-dimensional $\mathbb R$-vector space $V$, and $Q$ is a polyhedron in a finite-dimensional $\mathbb R$-vector space $W$, then $\left\lbrace f\in\mathrm{Hom}_{\mathbb R}\left(V,W\right) \mid f\left(P\right) \subseteq Q\right\rbrace$ is a polyhedron in $\mathrm{Hom}_{\mathbb R}\left(V,W\right)$. (This is inspired by Definition 9.16 in Günter M. Ziegler, *Lectures on Polytopes*, 1995.)

**Question:**

**6.** If $P$ is a polyhedron in a finite-dimensional $\mathbb R$-vector space $V$, and $Q$ is a polyhedron in a finite-dimensional $\mathbb R$-vector space $W$, then is it true that the convex hull of the set $\left\lbrace p \otimes q \mid p\in P,\ q\in Q \right\rbrace $ is a polyhedron in $V\otimes W$ ?

This holds for polytopes, and follows in that case from §2.5 of Lawrence Valby, *A Category of Polytopes* (caveat: my convex hull is not his $P\otimes Q$, but rather the image of his $P\otimes Q$ under a surjection which keeps the $p_iq_j$ coordinates and forgets the $p_i$, $q_j$ and $1$ coordinates); but the argument there does not generalize to polyhedra. On the other hand, I am at a loss when trying to find a counterexample. Any ideas?

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