description/proof of that for simplicial complex, point on underlying space is on simplex interior of unique simplex
Topics
About: vectors space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
- 4: Proof
Starting Context
- The reader knows a definition of simplicial complex.
- The reader knows a definition of simplex interior of affine simplex.
Target Context
- The reader will have a description and a proof of the proposition that for any simplicial complex, any point on the underlying space is on the simplex interior of a unique simplex.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(V\): \(\in \{\text{ the real vectors spaces }\}\)
\(C\): \(\in \{\text{ the simplicial complexes on } V\}\)
\(\vert C \vert\): \(= \text{ the underlying space of } C\)
\(p\): \(\in \vert C \vert\)
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Statements:
\(\exists ! S \in C (p \in S^\circ)\), where \(!\) denotes unique existence and \(\circ\) denotes the simplex interior
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2: Natural Language Description
For any real vectors space, \(V\), any simplicial complex, \(C\), on \(V\), the underlying space, \(\vert C \vert\), and any point, \(p \in \vert C \vert\), there is a unique simplex, \(S \in C\), such that \(p \in S^\circ\).
3: Note
The simplex interior of any vertex, which is a 0-simplex, is the set that consists of only the vertex. So, "is on the simplex interior of a unique simplex" may mean being a vertex.
4: Proof
\(p \in \vert C \vert\) implies that \(p \in [p_0, ..., p_n]\) for a \([p_0, ..., p_n] \in C\) (there may be some multiple such simplexes but pick up any one of them). \(p = \sum_{j \in \{0, ..., n\}} t^j p_j\). As \(\sum_{j \in \{0, ..., n\}} t^j = 1\) and \(0 \le t^j\), there is the subset, \(J = \{j \in \{0, ..., n\} \vert 0 \lt t^j\} = \{j_0, ..., j_k\} \subseteq\{0, ..., n\}\). \(S := [p_{j_0}, ..., p_{j_k}] \in C\), and \(p \in S^\circ\). So, \(p\) is on the simplex interior of at least 1 simplex, \(S\).
Let us suppose that \(p\) is also on the simplex interior of an \(S' \in C\).
\(p \in S^\circ \cap S'^\circ \subseteq S \cap S'\). \(S \cap S'\) is a face of \(S\). But if \(S \cap S'\) was a proper face of \(S\), \(p \in S \cap S'\) would not be on the simplex interior of \(S\): the simplex interior, \(S^\circ\), is \(S\) minus all the proper faces. So, \(S \cap S' = S\). Likewise (by symmetry), \(S \cap S' = S'\). So, \(S = S'\).
So, \(p\) is on the interior of the unique simplex, \(S\).
