# Characterizing f-vectors of regular(Delaunay) triangulations?

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:

2. Macaulay's theorem (about shellable simplicial complexes)

3. 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.

• It would be a (pleasant) surprise if anything even close to what you are asking about was already known. For instance, the following PhD thesis from a few years ago implies that such a characterization is unknown even for $5$-dimensional triangulated balls. See arxiv.org/abs/0912.2091 – Vidit Nanda Mar 4 '13 at 5:50
• Thank you so much indeed. Your link and the references in that article gave me a valuable source for more searching and I believe I'm very close to a kind of answer I was looking for. Thanks! – Daniel J. Park Mar 4 '13 at 13:03
• Another reference which looks fairly related is Ziegler's paper "Shelling polyhedral 3-balls and 4-polytopes". (Likely you've already seen this, although you don't mention it in your question.) He's mainly looking at extendable shellability in there, however. – Russ Woodroofe Mar 4 '13 at 14:47