definition of simplex boundary of affine 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
Starting Context
- The reader knows a definition of simplex interior of affine simplex.
Target Context
- The reader will have a definition of simplex boundary of affine 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:
\( V\): \(\in \{\text{ the real vectors spaces }\}\)
\( \{p_0, ..., p_n\}\): \(\subseteq V\), \(\in \{\text{ the affine-independent sets of base points on } V\}\)
\( [p_0, ..., p_n]\): \(= \{\sum_{j = 0 \sim n} t^j p_j \in V \vert t^j \in \mathbb{R}, \sum_{j = 0 \sim n} t^j = 1 \land 0 \le t^j\}\)
\( [p_0, ..., p_n]^\circ\): \(= \text{ the simplex interior of } [p_0, ..., p_n]\)
\(*bou [p_0, ..., p_n]\): \(= [p_0, ..., p_n] \setminus [p_0, ..., p_n]^\circ\)
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Conditions:
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2: Natural Language Description
For any real vectors space, \(V\), any affine-independent set of base points on \(V\), \(\{p_0, ..., p_n\}\), the affine simplex, \([p_0, ..., p_n] = \{\sum_{j = 0 \sim n} t^j p_j \in V \vert t^j \in \mathbb{R}, \sum_{j = 0 \sim n} t^j = 1 \land 0 \le t^j\}\), and the simplex interior, \([p_0, ..., p_n]^\circ\), of \([p_0, ..., p_n]\), \([p_0, ..., p_n] \setminus [p_0, ..., p_n]^\circ\), denoted as \(bou [p_0, ..., p_n]\)
3: Note
The reason why it is qualified as "simplex boundary" is that it can be different from 'topological boundary': when a simplicial complex consists of a simplex and its proper faces, the topological boundary of the simplex on the underlying space of the complex is the empty set, because the complement of the simplex is empty on the underlying space; in the same case, the topological boundary of any proper face of the simplex on the underlying space of the complex is the whole face.
Prevalently, the simplex boundary of \(S\) is denoted as \(\dot{S}\), which is in fact fine in that case, but this definition used \(bou [p_0, ..., p_n]\) because \(\dot{[p_0, ..., p_n]}\) seems somewhat too unnoticealble (if not, \(\dot{[p_0, ..., p_n]}\) is perfectly OK).
