I have started studying regular subdivisions of the $n$-cube, and came across the following post: Regularity of Delaunay triangulation of a hypercube.

My question is whether the "standard triangulation" of the $n$-cube described in the answer of that post is always regular. I know that the answer is yes in two and three dimensions, and, intuitively, I would guess that it is also true for arbitrary $n$. I'd be grateful for any help.

(A subdivision is regular if it can be obtained by lifting the vertices to $\mathbb{R}^{n+1}$, and projecting the lower faces of the convex hull of the lifted vertices to $\mathbb{R}^n$.)

  • 1
    $\begingroup$ More generally the canonical triangulation of the order polytope of any poset (with simplices corresponding to linear extensions) is regular. But this question seems a little too low-level for this site. $\endgroup$ Feb 25, 2020 at 21:41
  • $\begingroup$ See pp. 439-440 of csun.edu/~ctoth/Handbook/chap16.pdf. $\endgroup$ Feb 25, 2020 at 21:43
  • $\begingroup$ @SamHopkins Thank you! I‘m new to this topic, and wasn‘t aware that it’s that simple. $\endgroup$
    – KTree
    Feb 26, 2020 at 7:40


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