# 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

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.