# How many $k$-simplices does a non-boundary $(k-1)$-simplex lie in for a triangulation of a $k$-simplex?

A $$k$$-simplex is defined as the convex hull of $$k+1$$ affinely independent vectors (vertices). A face of a simplex is a convex hull of a subset of the vertices.

A triangulation of a simplex $$X$$ is a collection of simplices such that the union of them is $$X$$ and the intersection of any two of them is a face of each of the two simplices.

For a triangulation of $$\sigma_k:=\{(x_1,...,x_{k+1}):x_1+\cdots+x_{k+1}=1, x_i\ge 0\}$$ and a non-boundary $$(k-1)$$-simplex $$A$$ in the triangulation, i.e., $$A$$ is not contained in any boundary of $$\sigma_k$$, I am wondering how many $$k$$-simplices in the triangulation does $$A$$ lie in?

And is there any good reference to advance understanding of the simplices?

I think the answer is 2 but I don't know how to prove it rigorous: consider the hyperplane through the vertices $$v_1,...,v_k$$ of $$A$$ in $$\mathbb{R}^k$$. Then some (non-zero many) vertices of the triangulation should be on one side and some (non-zero many) on the other side, as $$A$$ is not on the boundary of $$\sigma_k$$. Since the simplices in a triangulation only intersect at faces, there should be only two $$k$$-simplices sit on $$A$$, otherwise two of them sit on the same side of the hyperplane and their intersection is larger than $$A$$ that is in the hyperplane.

• What have you tried? Just saying, "I think this is answer, but don't know what to do" is not enough. May 30, 2022 at 4:05
• Suggestion for reference in the subject of "simplex" is more welcome. May 30, 2022 at 10:41

Yes, your intuition is right. Work in the $$k$$-dimensional affine space$$~S$$ spanned by $$\sigma_k$$ (so $$S=\{\,(x_1,\ldots,x_{k+1})\mid x_1+\cdots+x_{k+1}=0\,\}$$). For any $$k$$-simplex$$~\Delta$$ (of the triangulation), any face $$F$$ of $$\Delta$$ (a $$k-1$$ simplex of the triangulation), and any interior point $$p$$ of $$F$$, it is the case that any sphere in $$S$$ centred in $$p$$ is cut in two halves by the affine hyperplane $$H$$ of $$S$$ spanned by $$F$$. Moreover if the sphere is chosen sufficiently small, one of those two halves is contained in$$~\Delta$$, while the interior of the other does not meet$$~\Delta$$. If $$F$$ is not part of a face of $$\sigma_k$$, then the entire sphere (if sufficiently small) is contained in $$\sigma_k$$, so there must be another $$k$$-simplex$$~\Delta'$$ of the triangulation that has $$p$$ on its boundary and contains (part of) the interior of the half-sphere that did not meet$$~\Delta$$. But then, again with a sufficiently small sphere, it contains all of that half-sphere. There cannot be more $$k$$-simplices of the triangulation that have $$p$$ on their boundary, as they would have to have interior points in common with either $$\Delta$$ or $$\Delta'$$, which is forbidden.