632: For Simplicial Complex, Point on Underlying Space Is on Simplex Interior of Unique Simplex

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description/proof of that for simplicial complex, point on underlying space is on simplex interior of unique simplex

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Topics

About: vectors space

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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

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Starting Context

• The reader knows a definition of simplicial complex.
• The reader knows a definition of simplex interior of affine simplex.

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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.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$V$: $\in \{\text{ the real vectors spaces }\}$
$C$: $\in \{\text{ the simplicial complexes on }V\}$
$|C|$: $=\text{ the underlying space of }C$
$p$: $\in |C|$
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Statements:
$\mathrm{\exists }!S\in C(p\in S^{\circ })$, 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, $|C|$, and any point, $p\in |C|$, there is a unique simplex, $S\in C$, such that $p\in S^{\circ }$.

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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.

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4: Proof

$p\in |C|$ 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, $J=\{j\in \{0,...,n\}|0<t^j\}=\{j_0,...,j_k\}\subseteq \{0,...,n\}$. $S:=[p_{j_0},...,p_{j_k}]\in C$, 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 $S^{\prime }\in C$.

$p\in S^{\circ }\cap S^{\prime \circ }\subseteq S\cap S^{\prime }$. $S\cap S^{\prime }$ is a face of $S$. But if $S\cap S^{\prime }$ was a proper face of $S$, $p\in S\cap S^{\prime }$ would not be on the simplex interior of $S$: the simplex interior, $S^{\circ }$, is $S$ minus all the proper faces. So, $S\cap S^{\prime }=S$. Likewise (by symmetry), $S\cap S^{\prime }=S^{\prime }$. So, $S=S^{\prime }$.

So, $p$ is on the interior of the unique simplex, $S$.

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References

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