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As pointed out by Sam Hopkins in a comment above, secondary polytopes can be seen as a particular case of the fiber polytopes of Billera and Sturmfels (https://doi.org/10.2307/2946575). This fiber polytope view provides the answer (or, at least, one answer) to what each point in the secondary polytope represents:

Consider a point configuration $A=\{a_1,\dots,a_n\}\subset \mathbb R^d$ (For the secondary polytope of a polytope $P$, let $A$ be the set of vertices of $P$, and substitute $P$ for $\operatorname{conv(A)}$ in all the description below). There is a canonical projection \begin{align} \pi: \Delta^{n-1} &\to \operatorname{conv(A)}\\ e_i &\mapsto a_i, \end{align} where $e_1,\dots,e_n$ are the standard basis in $\mathbb R^n$ and $\Delta^{n-1}$ is their convex hull, that is, the standard $n-1$-simplex.

Now, consider the space $\mathcal S$ of all sections of $\pi$. That is, elements of $\mathcal S$ are maps $s:\operatorname{conv(A)} \to \Delta^{n-1}$ such that $\phi\circ s$ is the identity map. Put differently, they are maps that choose a point in the fiber $\pi^{-1}(x)$ for each point $x\in \operatorname{conv(A)}$. (You can restrict to continuous sections, or to piecewise-linear sections, but you don't need to; what I say below holds for arbitrary sections too, as long as you can integrate them to compute their average).

Finally, consider the map that computes the average (or barycenter) of each such section. This is a map from $\mathcal S$ to the fiber $\pi^{-1}(b)$, where $b$ is the barycenter of $\operatorname{conv(A)}$. The secondary polytope of $\operatorname{conv(A)}$ is nothing but the image of this last map.

That is, each point in the secondary polytope corresponds, not uniquely, to a section of $\pi$; or, more precisely, it corresponds to the set of all sections with that average.

Vertices of the secondary polytope are special in that each of them is the image of a unique, and extremal, section, obtained by ``coherently'' picking the vertex of each fiber in a fixed direction. The section obtained in this coherent way is a union of faces of $\Delta^{n-1}$ forming (via $\pi$) the regular triangulation of $A$ corresponding to that vertex. This is why regular triangulations are also sometimes called coherent.

As pointed out by Sam Hopkins in a comment above, secondary polytopes can be seen as a particular case of the fiber polytopes of Billera and Sturmfels (https://doi.org/10.2307/2946575). This fiber polytope view provides the answer (or, at least, one answer) to what each point in the secondary polytope represents:

Consider a point configuration $A=\{a_1,\dots,a_n\}\subset \mathbb R^d$ (For the secondary polytope of a polytope $P$, take as $A$ the set of vertices of $P$, and substitute $P$ for $\operatorname{conv(A)}$ in all the description below). There is a canonical projection \begin{align} \pi: \Delta^{n-1} &\to \operatorname{conv(A)}\\ e_i &\mapsto a_i, \end{align} where $e_1,\dots,e_n$ are the standard basis in $\mathbb R^n$ and $\Delta^{n-1}$ is their convex hull, that is, the standard $n-1$-simplex.

Now, consider the space $\mathcal S$ of all sections of $\pi$. That is, elements of $\mathcal S$ are maps $s:\operatorname{conv(A)} \to \Delta^{n-1}$ such that $\phi\circ s$ is the identity map. Put differently, they are maps that choose a point in the fiber $\pi^{-1}(x)$ for each point $x\in \operatorname{conv(A)}$. (You can restrict to continuous sections, or to piecewise-linear sections, but you don't need to; what I say below holds for arbitrary sections too, as long as you can integrate them to compute their average).

Finally, consider the map that computes the average (or barycenter) of each such section. This is a map from $\mathcal S$ to the fiber $\pi^{-1}(b)$, where $b$ is the barycenter of $\operatorname{conv(A)}$. The secondary polytope of $\operatorname{conv(A)}$ is nothing but the image of this last map.

That is, each point in the secondary polytope corresponds, not uniquely, to a section of $\pi$; or, more precisely, it corresponds to the set of all sections with that average.

Vertices of the secondary polytope are special in that each of them is the image of a unique, and extremal, section, obtained by ``coherently'' picking the vertex of each fiber in a fixed direction. The section obtained in this coherent way is a union of faces of $\Delta^{n-1}$ forming (via $\pi$) the regular triangulation of $A$ corresponding to that vertex. This is why regular triangulations are also sometimes called coherent.

As pointed out by Sam Hopkins in a comment above, secondary polytopes can be seen as a particular case of the fiber polytopes of Billera and Sturmfels (https://doi.org/10.2307/2946575). This fiber polytope view provides the answer (or, at least, one answer) to what each point in the secondary polytope represents:

Consider a point configuration $A=\{a_1,\dots,a_n\}\subset \mathbb R^d$ (For the secondary polytope of a polytope $P$, let $A$ be the set of vertices of $P$). There is a canonical projection \begin{align} \phi: \Delta^{n-1} &\to \operatorname{conv(A)}\\ e_i &\mapsto a_i, \end{align} where $e_1,\dots,e_n$ are the standard basis in $\mathbb R^n$ and $\Delta^{n-1}$ is their convex hull, that is, the standard $n-1$-simplex.

Now, consider the space $\mathcal S$ of all sections of $\pi$. That is, elements of $\mathcal S$ are maps $s:\operatorname{conv(A)} \to \Delta^{n-1}$ such that $\phi\circ s$ is the identity map. Put differently, they are maps that choose a point in the fiber $\phi^{-1}(x)$ for each point $x\in \operatorname{conv(A)}$. (You can restrict to continuous sections, or to piecewise-linear sections, but you don't need to; what I say below holds for arbitrary sections too, as long as you can integrate them to compute their average).

Now consider the map that computes the average (or barycenter) of each such section. This is a map from $\mathcal S$ to the fiber $\phi^{-1}(b)$, where $b$ is the barycenter of $\operatorname{conv(A)}$. The secondary polytope of $\operatorname{conv(A)}$ is nothing but the image of this last map.

That is, each point in the secondary polytope corresponds, not uniquely, to a section of $\phi$; or, more precisely, it corresponds to the set of all sections with that average.

Vertices of the secondary polytope are special in that each of them is the image of a unique, and extremal, section, obtained by ``coherently'' picking the vertex of each fiber in a fixed direction. The section obtained in this coherent way is a union of faces of $\Delta^{n-1}$ forming (via $\phi$) the regular triangulation of $A$ corresponding to that vertex. This is why regular triangulations are also sometimes called coherent.

As pointed out by Sam Hopkins in a comment above, secondary polytopes can be seen as a particular case of the fiber polytopes of Billera and Sturmfels (https://doi.org/10.2307/2946575). This fiber polytope view provides the answer (or, at least, one answer) to what each point in the secondary polytope represents:

Consider a point configuration $A=\{a_1,\dots,a_n\}\subset \mathbb R^d$ (For the secondary polytope of a polytope $P$, let $A$ be the set of vertices of $P$, and substitute $P$ for $\operatorname{conv(A)}$ in all the description below). There is a canonical projection \begin{align} \pi: \Delta^{n-1} &\to \operatorname{conv(A)}\\ e_i &\mapsto a_i, \end{align} where $e_1,\dots,e_n$ are the standard basis in $\mathbb R^n$ and $\Delta^{n-1}$ is their convex hull, that is, the standard $n-1$-simplex.

Now, consider the space $\mathcal S$ of all sections of $\pi$. That is, elements of $\mathcal S$ are maps $s:\operatorname{conv(A)} \to \Delta^{n-1}$ such that $\phi\circ s$ is the identity map. Put differently, they are maps that choose a point in the fiber $\pi^{-1}(x)$ for each point $x\in \operatorname{conv(A)}$. (You can restrict to continuous sections, or to piecewise-linear sections, but you don't need to; what I say below holds for arbitrary sections too, as long as you can integrate them to compute their average).

Finally, consider the map that computes the average (or barycenter) of each such section. This is a map from $\mathcal S$ to the fiber $\pi^{-1}(b)$, where $b$ is the barycenter of $\operatorname{conv(A)}$. The secondary polytope of $\operatorname{conv(A)}$ is nothing but the image of this last map.

That is, each point in the secondary polytope corresponds, not uniquely, to a section of $\pi$; or, more precisely, it corresponds to the set of all sections with that average.

Vertices of the secondary polytope are special in that each of them is the image of a unique, and extremal, section, obtained by ``coherently'' picking the vertex of each fiber in a fixed direction. The section obtained in this coherent way is a union of faces of $\Delta^{n-1}$ forming (via $\pi$) the regular triangulation of $A$ corresponding to that vertex. This is why regular triangulations are also sometimes called coherent.

As pointed out by Sam Hopkins in a comment above, secondary polytopes can be seen as a particular case of the fiber polytopes of Billera and Sturmfels (https://doi.org/10.2307/2946575). This fiber polytope view provides the answer (or, at least, one answer) to what each point in the secondary polytope represents:

A point in the secondary polytope of a point configuration $A=\{a_1,\dots,a_n\}$ corresponds (not uniquely) to a section of the natural projection \begin{align} \phi: \Delta^{n-1} &\to \operatorname{conv(A)}\\ e_i &\mapsto a_i, \end{align} where $e_1,\dots,e_n$ are the standard basis in $\mathbb R^n$ and $\Delta^{n-1}$ is their convex hull, that is, the standard $n-1$-simplex. More precisely, the projection map $\phi$ induces a (non-surjective) map from the set of all sections to $\operatorname{conv(A)}$, sending each section to its average, and the secondary polytope is the image of this map (perhaps scaled, depending on your choice of normalisation).

Vertices of the secondary polytope are special in that each of them is the image of a unique, and extremal, section, obtained by ``coherently'' picking the vertex of each fiber in a fixed direction. The section obtained in this extremal case is a union of faces of $\Delta^{n-1}$ forming (via $\phi$) the regular triangulation of $A$ corresponding to that vertex. This is why regular triangulations are also sometimes called coherent.

Note: I am writing things in the language of point configurations. For a polytope $P$ as in the original post, the point configuration $A$ is the vertex set of $P$.

As pointed out by Sam Hopkins in a comment above, secondary polytopes can be seen as a particular case of the fiber polytopes of Billera and Sturmfels (https://doi.org/10.2307/2946575). This fiber polytope view provides the answer (or, at least, one answer) to what each point in the secondary polytope represents:

Consider a point configuration $A=\{a_1,\dots,a_n\}\subset \mathbb R^d$ (For the secondary polytope of a polytope $P$, let $A$ be the set of vertices of $P$). There is a canonical projection \begin{align} \phi: \Delta^{n-1} &\to \operatorname{conv(A)}\\ e_i &\mapsto a_i, \end{align} where $e_1,\dots,e_n$ are the standard basis in $\mathbb R^n$ and $\Delta^{n-1}$ is their convex hull, that is, the standard $n-1$-simplex.

Now, consider the space $\mathcal S$ of all sections of $\pi$. That is, elements of $\mathcal S$ are maps $s:\operatorname{conv(A)} \to \Delta^{n-1}$ such that $\phi\circ s$ is the identity map. Put differently, they are maps that choose a point in the fiber $\phi^{-1}(x)$ for each point $x\in \operatorname{conv(A)}$. (You can restrict to continuous sections, or to piecewise-linear sections, but you don't need to; what I say below holds for arbitrary sections too, as long as you can integrate them to compute their average).

Now consider the map that computes the average (or barycenter) of each such section. This is a map from $\mathcal S$ to the fiber $\phi^{-1}(b)$, where $b$ is the barycenter of $\operatorname{conv(A)}$. The secondary polytope of $\operatorname{conv(A)}$ is nothing but the image of this last map.

That is, each point in the secondary polytope corresponds, not uniquely, to a section of $\phi$; or, more precisely, it corresponds to the set of all sections with that average.

Vertices of the secondary polytope are special in that each of them is the image of a unique, and extremal, section, obtained by ``coherently'' picking the vertex of each fiber in a fixed direction. The section obtained in this coherent way is a union of faces of $\Delta^{n-1}$ forming (via $\phi$) the regular triangulation of $A$ corresponding to that vertex. This is why regular triangulations are also sometimes called coherent.