The question is this; *is there a complete characterization of all the f-vectors of regular triangulations?*

(other names for this can be: Delaunay triangulation, coherent triangulation, convex triangulation, Gale triangulation, or something else. It means the lower faces of a simplicial polytope.)

A brief history is this. For each polytopal complex $\mathcal{C}$, its *f*-vector is defined by $\overrightarrow{f}(\mathcal{C}) = (f_{-1},f_0,f_1,\cdots,f_{d-1})$ where $f_i$ is the number of $i$-faces of $\mathcal{C}$. An auxiliary vector $\overrightarrow{h}(\mathcal{C}) = (h_0,h_1,\cdots,h_d)$ which is a linear transform of $\overrightarrow{f}(\mathcal{C})$ carries the same information as $\overrightarrow{f}(\mathcal{C})$.

The Euler-Poincare formula is the only nontrivial linear equation that is satisfied by all *f*-vectors of the boundary complexes $\partial\mathcal{P}$ of $d$-polytopes $\mathcal{P}$ in general. If we confine ourselves to simplicial $d$-polytopes only, the famous Dehn-Sommerville equations are satisfied, which can be formulated in terms of *h*-vectors as
\begin{equation*}
h_k = h_{d-k} \text{ for $k=0,1,\cdots,d$.}
\end{equation*}

An immediate question is about its converse, i.e., characterizing all the possible *f*-vectors. There are famous results in this direction:

Kruskal-Katona theorem(about simplicial complexes)

Macaulay's theorem (about shellable simplicial complexes)

The

*g*-theorem by Billera, Lee and Stanley (about simplicial polytopes)

The first two theorems say that any vector $\overrightarrow{a}$ can be the *f*-vector of some shellable simplicial complex whenever the sizes of $a_{-1},a_0,\cdots,a_{d-1}$ are 'reasonable'. ('reasonable' means that the *h*-vector should be an M-sequence or O-sequence.) The third one basically says that the size constraint(being M-sequence) and Dehn-Sommerville equations characterize the *f*-vectors of (boundaries of) simplicial polytopes completely. Therefore, in an expository manner we may write
\begin{equation*}
\frac{\{\text{shellable simplicial complexes} \}}{\partial(\{ \text{simplicial polytopes} \})} \sim \text{Dehn-Sommerville}.
\end{equation*}

My question is the analogue of this;

\begin{equation*}
\frac{\{\text{shellable simplicial complexes} \}}{\{ \text{regular triangulations} \}} \sim \;?
\end{equation*}

I've been searching for this, but nothing in this direction appeared to me.