First, let's consider an almost trivial notion. With any subspace $V\subset \mathbb R^n$ we associate a convex polytope $P(V)\subset V^*$ as follows. Each of the $n$ coordinates in $\mathbb R^n$ is a functional on $\mathbb R^n$ and, subsequently, on $V$. This provides $n$ elements of $V^*$ and we let $P(V)$ be their convex hull. One may also consider orthogonal projections of the $n$ basis vectors onto $V$ with respect to the standard scalar product. The convex hull of the resulting points will be equivalent to $P(V)$ but I will stick to the dual language.

On just a slightly less trivial note, for a polyhedral fan $\mathcal F$ in $\mathbb R^n$ and a cone $C\in \mathcal F$ denote $P(C)=P(V)$ where $V$ is the linear span of $C$. Evidently, for a face $D$ of $C$ we have a natural projection of $P(C)$ onto $P(D)$. Together these maps form a "projective system" over the face lattice of $\mathcal F$.

Now, I claim that there are two (related) contexts in which this notion seems at least somewhat interesting: secondary fans and Gröbner fans. Below I name some of the properties which motivate this claim but overall I feel like these ideas should be present in the literature and, perhaps, there's some more natural language/context for this which I'm not seeing. So any references or alternative approaches are welcome.

Let $Q$ be a convex polytope with $n$ vertices and let $\mathcal F$ be the secondary fan of its vertex set in $\mathbb R^n$. Then we have the following properties.

• For $C_0$ the minimal cone in $\mathcal F$ the polytope $P(C_0)$ is (linearly equivalent to) $Q$.
• All polytopes $P(C)$ have exactly $n$ vertices. $P(C)$ is an $(n-1)$-dimensional simplex for a maximal cone $C$.
• Every cone $C$ corresponds to a regular subdivision $\Delta$ or $Q$. Via the identification between $Q$ and $P(C_0)$ we may view $\Delta$ as subdivision of $P(C_0)$. Consider the projection $\pi:P(C)\to P(C_0)$. Then $\Delta$ is induced by $\pi$ meaning that for every part $A$ of $\Delta$ there's a (unique) face of $P(C)$ which is mapped bijectively to $A$ by $\pi$.

Next, let $I\subset R=\mathbb C[x_1,\dots,x_n]$ be a homogeneous prime monomial-free ideal and $\mathcal F$ be its Gröbner fan in $\mathbb R^n$, then we have the following.

• Let $T\subset(\mathbb C^*)^n$ be the maximal subtorus of the coordinate torus preserving $I$. Let $C_0$ be the minimal cone in $\mathcal F$. Then $\dim C_0=\dim T$ and for a generic point $x\in \mathrm{Proj}(R/I)$ the orbit closure $\overline{Tx}$ is the toric variety of $P(C_0)$.
• Suppose a cone $C$ in $\mathcal F$ is maximal in the tropical subfan and corresponds to a prime initial ideal $J$. This implies that $J$ is toric and $\mathrm{Proj}(R/J)$ is a flat toric degeneration of $\mathrm{Proj}(R/I)$. In fact, it is the toric variety of $P(C)$.