# Definition of and intuition for regular subdivisions of a polytope

I'm doing a research project that involves subdividing a product of simplices. Specifically, I'm looking at theorem 2.4 from this paper:

math.sfsu.edu/federico/Articles/tropOMs.pdf

which references for proof:

www.emis.de/journals/DMJDMV/vol-09/01.pdf

Neither paper defines "regular subdivision" or references a definition, and other literature definitions are rare and not obviously equivalent. Can someone with more background in this field give a formal definition, and if that doesn't by itself give a reasonable intuition, some examples of regular vs. irregular subdivisions?

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You will find a definition of coherent subdivisions in Discriminants, Resultants, and Multidimensional Determinants by I. M. Gelfand, M. Kapranov and A. Zelevinsky, which I think is the same as your regular. –  Mariano Suárez-Alvarez Jul 8 '10 at 2:19
Sturmfels's Gröbner bases and convex polytopes is another very good reference. –  Mariano Suárez-Alvarez Jul 8 '10 at 2:23

## 2 Answers

A very good reference for this is the recent book of De Loera, Rambau and Stantos called "Triangulations: Structures for Algorithms and Applications". They have a whole chapter on regular triangulations. I don't know if the book is in print yet. It's possible that careful googling could reveal drafts of the book that are still online.

The classic example of a triangulation of a point set that is not regular is the following: Draw two concentric triangles with vertices {1,2,6} and {3,4,5}. The following is a triangulation: {16,26,21, 34,45,43, 36,56, 13,14, 24,25}, where {ij} means draw an edge from i to j.

You can see that this is not regular by assuming that {3,4,5} were not lifted above the plane, and then trying to lift vertices, 1,2,6 to get the remaining faces. You will find that you need the heights to satisfy height(1) < height(2) < height(6) < height(1). This is spelled out in complete detail in Sturmfels' book.

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I did not look at your references, so apologies if this is irrelevant. Carl Lee defines a regular subdivision of a set of points $V$ (in "Subdivisions and triangulations of polytopes," Handbook of Discrete and Computational Geometry, p.387) as one which is obtained by regarding $V$ as in $\mathbb{R}^d$, and projecting the lower facets of the convex hull of $$(v_1, \alpha_1), \ldots, (v_n, \alpha_n)$$ where the $\alpha$'s are arbitrary real numbers. So essentially the regular subdivisions are projections from $\mathbb{R}^{d+1}$ to $\mathbb{R}^d$. All regular subdivisions are shellable, among their other nice properties they enjoy.

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