Understanding convex polytopes (I)

This summer, i'm working in an intense month-long research program about discrete geometry (a.k.a. combinatorial geometry). To keep track of what i do, what i learn and what i don't, i'll be writing (hopefully) regular posts regarding the work i do through the day. That should help me a ton, and hopefully someone out there as well?

Consider a set of points $P \subset \mathbb{R}^n$. We say that $P$ is convex if for any $x,y \in P$, the segment $[x,y]$ is contained in $P$. An intersection of convex sets is convex, so we define the convex hull of $P$ as the smallest convex set that contains $P$:

$$\text{ConvexHull}(P) := \bigcap \{P' \subset \mathbb{R}^n : P \subset P', \text{ for $P'$ convex}\}$$

One of the fundamental concepts is that of a polytope: there are two equivalent definitions.

A $\mathcal{V}$-polytope is the convex hull of a finite set of points in $\mathbb{R}^n$.

An $\mathcal{H}$-polytope is the intersection of a finite set of closed halfspaces in $\mathbb{R}^n$ (that is; the region enclosed by a set of inequalities).

That these two definitions are equivalent is not trivial, but it has been proven, and it is not what I'm specially interested about.

What I'm interested in is the combinatorial structure of a given polytope (or family of polytopes). We compress all the combinatorial information of a polytope in three fundamental invariants: the $f$, $h$ and $\gamma$-vectors. The $f$-vector is defined as the vector $(f_0, f_1, ... , f_d)$, where $f_i$ denotes the amount of $i$-faces of $P$: $f_0$ is the amount vertices, $f_1$ the amount of edges, etc...

(TODO: $h$ and $\gamma$-vectors. Their relation to the $f$-vector.)

We define the $n$-simplex $\Delta^n$ as the convex hull of $\{e_i : i \in [1, ..., n]\}$, where $e_i$ is the $i$th standard basis vector of $\mathbb{R}^n$. Notice that $\Delta^n$ is affinely independent.

A $d$-polytope $P$ ($d$ dimensions) is simple if each of its vertices is incident to $d$ edges. A $d$-polytope is simplicial if all of its faces are simplices.

(TODO: More definitions, clearer exposition, give intuition. What a bad rhyme.)

Now we turn our attention to a particular way of constructing polytopes: from graphs. Consider a graph $G$ on the labeled vertices $[1, ..., n]$ which is connected (there is a path that connects any pair of vertices, but it needn't be direct). Define the connected building set as the set of all subsets of vertices of $G$, such that the induced subgraph is connected:

$$\mathcal{B}(G) := \{U \subset G : G|_U \text{ is connected}\}$$

The polytope derived from this is defined as the Minkowski sum

$$P_{\mathcal{B}(G)} = \sum_{U\in\mathcal{B}(G)} \Delta_U$$

of $U$-simplices $\Delta_U := ConvexHull(e_i : i \in U)$. We call $P$ a graph-associahedron, and in general, the $P_{\mathcal{B}}$ of some building set $\mathcal{B}$ is called the nestohedron.

It is obvious that the choice of graph will determine the resulting polytope. We will study the polytopes generated by particular kinds of graphs (cyclic, complete, path, stellar, etc...): particularly, we will work towards fully determining their combinatorial structure by finding the $f$, $h$ and $\gamma$ vectors.

Note we required $G$ to be connected, because disconnected elements (their simplex representation are points) have no effect on the Minkowski sum: a point added to anything is just a translation.

My doubts at the moment
So what I'm interested in understanding is how to combinatorially describe a graph-associahedra. The Minkowski sum of simplices gets hairy pretty quickly, and I feel like any insight into the structure of $P_{\mathcal{B}(G)}$ must come from the structure of $G$ itself. But how do I translate graph properties to geometric properties? I'll try to go over simple examples, and generalize from there.