description/proof of that for simplicial complex, intersection of 2 affine simplexes determined by subsequences of ascending sequences of barycenters of faces of elements of complex is affine simplex determined by intersection of subsequences
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
- 4: Note
Starting Context
- The reader knows a definition of simplicial complex.
- The reader knows a definition of ascending sequence of barycenters of faces of affine simplex.
- The reader admits the proposition that for any simplicial complex, the intersection of any 2 simplexes is the simplex determined by the intersection of the sets of the vertexes of the simplexes.
- The reader admits the proposition that for any affine simplex, its any ascending sequence of faces, and the set of the barycenters of the faces, any convex combination of any subset of the set of the barycenters is a convex combination with respect to the set of the vertexes of the affine simplex.
Target Context
- The reader will have a description and a proof of the proposition that for any simplicial complex, the nonempty intersection of the 2 affine simplexes determined by any subsequences of any ascending sequences of barycenters of faces of any elements of the complex is the affine simplex determined by the intersection of the subsequences.
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:
\(V\): \(\in \{\text{ the real vectors spaces }\}\)
\(C\): \(\in \{\text{ the affine complexes on } V\}\)
\([p_0, ..., p_n]\): \(\in C\)
\([q_0, ..., q_m]\): \(\in C\)
\((b_0, ..., b_n)\): \(= (bary ([p_{\sigma_0}]), bary ([p_{\sigma_0}, p_{\sigma_1}]), ..., bary ([p_{\sigma_0}, ..., p_{\sigma_n}]))\), \(\in \{\text{ the ascending sequences of barycenters of faces of } [p_0, ..., p_n]\}\)
\((c_0, ..., c_m)\): \(= (bary ([q_{\rho_0}]), bary ([q_{\rho_0}, q_{\rho_1}]), ..., bary ([q_{\rho_0}, ..., q_{\rho_m}]))\), \(\in \{\text{ the ascending sequences of barycenters of faces of } [q_0, ..., q_m]\}\)
\([b_{k_0}, ..., b_{k_l}]\): \(\{b_{k_0}, ..., b_{k_l}\} \subseteq \{b_0, ..., b_n\}\)
\([c_{i_0}, ..., c_{i_r}]\): \(\{c_{i_0}, ..., c_{i_r}\} \subseteq \{c_0, ..., c_m\}\)
\(S\): \(= [b_{k_0}, ..., b_{k_l}] \cap [c_{i_0}, ..., c_{i_r}]\)
//
Statements:
\(S \neq \emptyset\)
\(\implies\)
\(S = [\{b_{k_0}, ..., b_{k_l}\} \cap \{c_{i_0}, ..., c_{i_r}\}]\).
//
2: Natural Language Description
For any real vectors space, \(V\), any affine complex on \(V\), \(C\), any elements, \([p_0, ..., p_n], [q_0, ..., q_m] \in C\), any ascending sequences of barycenters of faces of \([p_0, ..., p_n]\) and \([q_0, ..., q_m]\), \((b_0, ..., b_n) := (bary ([p_{\sigma_0}]), bary ([p_{\sigma_0}, p_{\sigma_1}]), ..., bary ([p_{\sigma_0}, ..., p_{\sigma_n}]))\) and \((c_0, ..., c_m) := (bary ([q_{\rho_0}]), bary ([q_{\rho_0}, q_{\rho_1}]), ..., bary ([q_{\rho_0}, ..., q_{\rho_m}]))\), the affine simplexes determined by any subsequences, \([b_{k_0}, ..., b_{k_l}]\) where \(\{b_{k_0}, ..., b_{k_l}\} \subseteq \{b_0, ..., b_n\}\) and \([c_{i_0}, ..., c_{i_r}]\) where \(\{c_{i_0}, ..., c_{i_r}\} \subseteq \{c_0, ..., c_m\}\), and \(S = [b_{k_0}, ..., b_{k_l}] \cap [c_{i_0}, ..., c_{i_r}]\), if \(S \neq \emptyset\), \(S = [\{b_{k_0}, ..., b_{k_l}\} \cap \{c_{i_0}, ..., c_{i_r}\}]\).
3: Proof
\([p_0, ..., p_n] \cap [q_0, ..., q_m]\) is a face of \([p_0, ..., p_n]\) and a face of \([q_0, ..., q_m]\). Without loss of generality (just re-index the vertexes of \([p_0, ..., p_n]\) and \([q_0, ..., q_m]\) and change the permutations, \(\sigma\) and \(\rho\), accordingly), let \([p_0, ..., p_n] \cap [q_0, ..., q_m] = [p_0, ..., p_s]\) and \([q_0, ..., q_m] = [p_0, ..., p_s, q_{s + 1}, ..., q_m]\) (refer to the proposition that for any simplicial complex, the intersection of any 2 simplexes is the simplex determined by the intersection of the sets of the vertexes of the simplexes).
\([b_{k_0}, ..., b_{k_l}] \subseteq [p_0, ..., p_n]\) and \([c_{i_0}, ..., c_{i_r}] \subseteq [q_0, ..., q_m]\), by the proposition that for any affine simplex, its any ascending sequence of faces, and the set of the barycenters of the faces, any convex combination of any subset of the set of the barycenters is a convex combination with respect to the set of the vertexes of the affine simplex. So, \(S = [b_{k_0}, ..., b_{k_l}] \cap [c_{i_0}, ..., c_{i_r}] \subseteq [p_0, ..., p_n] \cap [q_0, ..., q_m] = [p_0, ..., p_s]\).
Let \(p \in S\) be any. \(p = \sum_{j \in \{0, ..., l\}} t^j b_{k_j} = \sum_{j \in \{0, ..., r\}} u^j c_{i_j}\), where \(\sum_{j \in \{0, ..., l\}} t^j = 1\) and \(0 \le t^j\) and \(\sum_{j \in \{0, ..., r\}} u^j = 1\) and \(0 \le u^j\).
Hereafter, we will profusely use the proposition that for any affine simplex, its any ascending sequence of faces, and the set of the barycenters of the faces, any convex combination of any subset of the set of the barycenters is a convex combination with respect to the set of the vertexes of the affine simplex for \(p\): \(p\) is a convex combination with respect to \(\{p_0, ..., p_n\}\) or \(\{q_0, ..., q_m\}\), which implies that for the expression with respect to \(\{p_0, ..., p_n\}\) or \(\{q_0, ..., q_m\}\), the coefficients are uniquely determined. Although we said "\(\{p_0, ..., p_n\}\) or \(\{q_0, ..., q_m\}\)", in fact, only the common \(\{p_0, ..., p_s\} \subseteq \{p_0, ..., p_n\}, \{q_0, ..., q_m\}\) really appears, as will be proved in the next paragraph.
Where \(0 \lt t^a\), \(b_{k_a}\) is the barycenter of a face of \([p_0, ..., p_s]\), because otherwise, the expression of \(p\) with respect to \(\{p_0, ..., p_n\}\) would contain \(p_t\) where \(s \lt t\) (supposing that the face contains \(p_t\), as any barycenter takes positive multiples of the vertexes and \(t^j\) s are all non-negative, the coefficient of \(p_t\) would never disappear), which would mean \(p \notin [p_0, ..., p_s]\), a contradiction. Likewise, where \(0 \lt u^a\), \(c_{i_a}\) is the barycenter of a face of \([p_0, ..., p_s]\).
Let us see only the barycenters that are effectively in \(p\) (whose coefficients are nonzero): let us take the subset, \(\{b'_0, ..., b'_a\} = \{b_{k_j} \in \{b_{k_0}, ..., b_{k_l}\} \vert 0 \lt t^j\}\), and the subset, \(\{c'_0, ..., c'_b\} = \{c_{i_j} \in \{c_{i_0}, ..., c_{i_r}\} \vert 0 \lt u^j\}\).
Let us note these facts: as \(\{b'_0, ..., b'_a\}\) or \(\{c'_0, ..., c'_b\}\) has been derived from the ascending sequence of faces, if any vertex, \(p_g\), first appears in a \(b'_h\) or \(c'_h\), \(p_g\) will keep appearing in all the succeeding \(b'_{h + 1}, ..., b'_a\) or \(c'_{h + 1}, ..., c'_b\); if any 2 vertexes, \(p_f, p_g\), first appear in the same \(b'_h\) or \(c'_h\), the coefficient of \(p_f\) and the coefficient of \(p_g\) will be the same (\(p_f\) and \(p_g\) appear always in the form, \(1 / d (... + p_f + ... + p_g + ...)\)); if any vertex, \(p_f\), appears earlier than any vertex, \(p_g\), the coefficient of \(p_f\) will be larger than the coefficient of \(p_g\) (after \(p_g\) has appeared, \(p_f\) and \(p_g\) appear always in the form, \(1 / d (... + p_f + ... + p_g + ...)\), while the earlier terms with only \(p_f\) remain.
If \(b'_0 \neq c'_0\), there would be a \(p_f\) that was in \(b'_0\) but not in \(c'_0\) without loss of generality. Then, \(p_f\) would have to appear in later \(c'_j\), which is impossible by the observation in the previous paragraph: if a common vertex, \(p_g\), was in \(b'_0\) and \(c'_0\), for \(\sum_{j \in \{0, ..., l\}} t^j b_{k_j}\), \(p_f\) and \(p_g\) would have the same coefficients, but for \(\sum_{j \in \{0, ..., r\}} u^j c_{i_j}\), \(p_f\) and \(p_g\) would have some different coefficients, impossible; if no common vertex was in \(b'_0\) and \(c'_0\), there would be a \(p_g\) that was in \(c'_0\) but not in \(b'_0\) (because \(c'_0\) had to have at least 1 vertex), then, \(p_g\) would have to appear in later \(b'_j\), but that would mean that for \(\sum_{j \in \{0, ..., l\}} t^j b_{k_j}\), \(p_f\) would have a larger coefficient than \(p_g\), while for \(\sum_{j \in \{0, ..., r\}} u^j c_{i_j}\), \(p_f\) would have a smaller coefficient than \(p_g\), impossible. So, \(b'_0 = c'_0\).
Likewise, \(b'_1 = c'_1\), because it is about adding some vertexes to \(b'_0 = c'_0\), and unless \(b'_1\) and \(c'_1\) get the same set of vertexes, a contradiction would occur. And so on. So, \(b'_j = c'_j\) until \(j\) reaches the smaller of \(a\) and \(b\). In fact, \(a = b\), because if \(a \lt b\), a vertex would appear only in \(\sum_{j \in \{0, ..., r\}} u^j c_{i_j}\) and if \(b \lt a\), a vertex would appear only in \(\sum_{j \in \{0, ..., l\}} t^j b_{k_j}\), impossible.
That means that for \(p = \sum_{j \in \{0, ..., l\}} t^j b_{k_j}\), where \(0 \lt t^j\), \(b_{k_j} \in \{c_{j_0}, ..., c_{j_r}\}\). So, \(S \subseteq [\{b_{k_0}, ..., b_{k_l}\} \cap \{c_{k_0}, ..., c_{k_r}\}]\).
On the other hand, \([\{b_{k_0}, ..., b_{k_l}\} \cap \{c_{k_0}, ..., c_{k_r}\}] \subseteq S\), because the left hand side is a face of \([b_{k_0}, ..., b_{k_l}]\) and a face of \([c_{k_0}, ..., c_{k_r}]\), which means that the left hand side is contained in \([b_{k_0}, ..., b_{k_l}]\) and is contained in \([c_{k_0}, ..., c_{k_r}]\), and so, is contained in the intersection.
So, \(S = [\{b_{k_0}, ..., b_{k_l}\} \cap \{c_{k_0}, ..., c_{k_r}\}]\).
4: Note
This proposition can be used for proving that the barycentric subdivision of any simplicial complex is indeed a simplicial complex.
