description/proof of that convex set spanned by non-affine-independent set of base points on real vectors space is not necessarily affine simplex spanned by affine-independent subset of base points
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: Proof
Starting Context
- The reader knows a definition of convex set spanned by possibly-non-affine-independent set of base points on real vectors space.
- The reader knows a definition of affine-independent set of points on real vectors space.
- The reader knows a definition of affine simplex.
Target Context
- The reader will have a description and a proof of the proposition that the convex set spanned by a non-affine-independent set of base points on a real vectors space is not necessarily any affine simplex spanned by an affine-independent subset of the base points.
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:
\(V\): \(\in \{\text{ the real vectors spaces }\}\)
\(\{p_0, ..., p_n\}\): \(\subseteq V\), \(\in \{\text{ the non-affine-independent sets of base points on } V\}\)
\(S\): \(= \{\sum_{j = 0 \sim n} t^j p_j \in V \vert t^j \in \mathbb{R}, \sum_{j = 0 \sim n} t^j = 1 \land 0 \le t^j\}\)
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Statements:
There may not be any \(J \subset \{0, ..., n\}\)
(
\(\{p_j \vert j \in J\} \in \{\text{ the affine-independent sets of base points on } V\}\)
\(\land\)
\(S = \{\sum_{j \in J} t^j p_j \in V \vert t^j \in \mathbb{R}, \sum_{j = 0 \sim n} t^j = 1 \land 0 \le t^j\}\)
)
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2: Natural Language Description
For a real vectors space, \(V\), and a non-affine-independent set of base points, \(\{p_0, ..., p_n\} \subseteq V\), the convex set spanned by the set of the base points, \(S := \{\sum_{j = 0 \sim n} t^j p_j \in V \vert t^j \in \mathbb{R}, \sum_{j = 0 \sim n} t^j = 1 \land 0 \le t^j\}\), is not necessarily any affine simplex spanned by an affine-independent subset of the base points.
3: Proof
A counterexample suffices. Let \(V = \mathbb{R}^2\), the Euclidean vectors space. Let \(\{p_0 = 0, p_1 = e_1, p_2 = e_2, p_3 = e_1 + e_2\} \subseteq V\), where \(\{e_1, e_2\}\) is the standard unit vectors for \(\mathbb{R}^2\), be the set of base points. While \(S\) is the unit square, any affine-independent subset (for example \(\{p_0, p_1, p_2\}\)) of the set of the base points does not span the square as the convex set (succinctly speaking, the square is not any affine simplex, and so, choosing any subset does not help).
