What is the proof for the statement about facets in a simplex triangulation?

[e(ho0n3]

First, some definitions:

An n-simplex is defined as the convex hull of n+1 affinely independent vectors in R^d (its vertices). A face of a simplex is defined to be the convex hull of any subset of its vertices. A facet of a n-simplex is a face that is an (n-1)-simplex. A triangulation T of an n-simplex S is a finite collection of n-simplices {S', S'', ...} (called the subsimplices of T) such that (i) their union equals S and (ii) the intersection of any two of them is either empty or a common face.

Now for the question:

Let S be a simplex and let T be a triangulation of S. Let F be a facet of a subsimplex S' of T. Then either F is contained in a facet of S or F is the facet of exactly one other simplex S''.

The question is, how does one prove the above statement? All the books I have checked consider this a trivial fact or leave it as an exercise to the reader. In trying to prove it, I was forced to use a lot of topological facts about simplices that I have not seen proved anywhere (e.g. a point in a simplex is an interior point if and only if its barycentric coordinates are all positive). Since the definition of a triangulation makes no use of any topology, it bugs me that I have had to use so much topology in trying to prove it. If anybody supplies me with a reference with a full proof, I would greatly appreciate it.

 

[lavinia]

First, some definitions:

An n-simplex is defined as the convex hull of n+1 affinely independent vectors in R^d (its vertices). A face of a simplex is defined to be the convex hull of any subset of its vertices. A facet of a n-simplex is a face that is an (n-1)-simplex. A triangulation T of an n-simplex S is a finite collection of n-simplices {S', S'', ...} (called the subsimplices of T) such that (i) their union equals S and (ii) the intersection of any two of them is either empty or a common face.

Now for the question:

Let S be a simplex and let T be a triangulation of S. Let F be a facet of a subsimplex S' of T. Then either F is contained in a facet of S or F is the facet of exactly one other simplex S''.

The question is, how does one prove the above statement? All the books I have checked consider this a trivial fact or leave it as an exercise to the reader. In trying to prove it, I was forced to use a lot of topological facts about simplices that I have not seen proved anywhere (e.g. a point in a simplex is an interior point if and only if its barycentric coordinates are all positive). Since the definition of a triangulation makes no use of any topology, it bugs me that I have had to use so much topology in trying to prove it. If anybody supplies me with a reference with a full proof, I would greatly appreciate it.

I think it follow immefiately from the definition of a triangulation.

 

[e(ho0n3]

I think it follow immefiately from the definition of a triangulation.

That's what many books say too, but I don't see how it immediately follows. For example, I don't see how it rules out that three subsimplices can share a common facet.