2024-06-09

622: 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

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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


  • 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: { the real vectors spaces }
C: { the affine complexes on V}
[p0,...,pn]: C
[q0,...,qm]: C
(b0,...,bn): =(bary([pσ0]),bary([pσ0,pσ1]),...,bary([pσ0,...,pσn])), { the ascending sequences of barycenters of faces of [p0,...,pn]}
(c0,...,cm): =(bary([qρ0]),bary([qρ0,qρ1]),...,bary([qρ0,...,qρm])), { the ascending sequences of barycenters of faces of [q0,...,qm]}
[bk0,...,bkl]: {bk0,...,bkl}{b0,...,bn}
[ci0,...,cir]: {ci0,...,cir}{c0,...,cm}
S: =[bk0,...,bkl][ci0,...,cir]
//

Statements:
S

S=[{bk0,...,bkl}{ci0,...,cir}].
//


2: Natural Language Description


For any real vectors space, V, any affine complex on V, C, any elements, [p0,...,pn],[q0,...,qm]C, any ascending sequences of barycenters of faces of [p0,...,pn] and [q0,...,qm], (b0,...,bn):=(bary([pσ0]),bary([pσ0,pσ1]),...,bary([pσ0,...,pσn])) and (c0,...,cm):=(bary([qρ0]),bary([qρ0,qρ1]),...,bary([qρ0,...,qρm])), the affine simplexes determined by any subsequences, [bk0,...,bkl] where {bk0,...,bkl}{b0,...,bn} and [ci0,...,cir] where {ci0,...,cir}{c0,...,cm}, and S=[bk0,...,bkl][ci0,...,cir], if S, S=[{bk0,...,bkl}{ci0,...,cir}].


3: Proof


[p0,...,pn][q0,...,qm] is a face of [p0,...,pn] and a face of [q0,...,qm]. Without loss of generality (just re-index the vertexes of [p0,...,pn] and [q0,...,qm] and change the permutations, σ and ρ, accordingly), let [p0,...,pn][q0,...,qm]=[p0,...,ps] and [q0,...,qm]=[p0,...,ps,qs+1,...,qm] (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).

[bk0,...,bkl][p0,...,pn] and [ci0,...,cir][q0,...,qm], 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=[bk0,...,bkl][ci0,...,cir][p0,...,pn][q0,...,qm]=[p0,...,ps].

Let pS be any. p=j{0,...,l}tjbkj=j{0,...,r}ujcij, where j{0,...,l}tj=1 and 0tj and j{0,...,r}uj=1 and 0uj.

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 {p0,...,pn} or {q0,...,qm}, which implies that for the expression with respect to {p0,...,pn} or {q0,...,qm}, the coefficients are uniquely determined. Although we said "{p0,...,pn} or {q0,...,qm}", in fact, only the common {p0,...,ps}{p0,...,pn},{q0,...,qm} really appears, as will be proved in the next paragraph.

Where 0<ta, bka is the barycenter of a face of [p0,...,ps], because otherwise, the expression of p with respect to {p0,...,pn} would contain pt where s<t (supposing that the face contains pt, as any barycenter takes positive multiples of the vertexes and tj s are all non-negative, the coefficient of pt would never disappear), which would mean p[p0,...,ps], a contradiction. Likewise, where 0<ua, cia is the barycenter of a face of [p0,...,ps].

Let us see only the barycenters that are effectively in p (whose coefficients are nonzero): let us take the subset, {b0,...,ba}={bkj{bk0,...,bkl}|0<tj}, and the subset, {c0,...,cb}={cij{ci0,...,cir}|0<uj}.

Let us note these facts: as {b0,...,ba} or {c0,...,cb} has been derived from the ascending sequence of faces, if any vertex, pg, first appears in a bh or ch, pg will keep appearing in all the succeeding bh+1,...,ba or ch+1,...,cb; if any 2 vertexes, pf,pg, first appear in the same bh or ch, the coefficient of pf and the coefficient of pg will be the same (pf and pg appear always in the form, 1/d(...+pf+...+pg+...)); if any vertex, pf, appears earlier than any vertex, pg, the coefficient of pf will be larger than the coefficient of pg (after pg has appeared, pf and pg appear always in the form, 1/d(...+pf+...+pg+...), while the earlier terms with only pf remain.

If b0c0, there would be a pf that was in b0 but not in c0 without loss of generality. Then, pf would have to appear in later cj, which is impossible by the observation in the previous paragraph: if a common vertex, pg, was in b0 and c0, for j{0,...,l}tjbkj, pf and pg would have the same coefficients, but for j{0,...,r}ujcij, pf and pg would have some different coefficients, impossible; if no common vertex was in b0 and c0, there would be a pg that was in c0 but not in b0 (because c0 had to have at least 1 vertex), then, pg would have to appear in later bj, but that would mean that for j{0,...,l}tjbkj, pf would have a larger coefficient than pg, while for j{0,...,r}ujcij, pf would have a smaller coefficient than pg, impossible. So, b0=c0.

Likewise, b1=c1, because it is about adding some vertexes to b0=c0, and unless b1 and c1 get the same set of vertexes, a contradiction would occur. And so on. So, bj=cj until j reaches the smaller of a and b. In fact, a=b, because if a<b, a vertex would appear only in j{0,...,r}ujcij and if b<a, a vertex would appear only in j{0,...,l}tjbkj, impossible.

That means that for p=j{0,...,l}tjbkj, where 0<tj, bkj{cj0,...,cjr}. So, S[{bk0,...,bkl}{ck0,...,ckr}].

On the other hand, [{bk0,...,bkl}{ck0,...,ckr}]S, because the left hand side is a face of [bk0,...,bkl] and a face of [ck0,...,ckr], which means that the left hand side is contained in [bk0,...,bkl] and is contained in [ck0,...,ckr], and so, is contained in the intersection.

So, S=[{bk0,...,bkl}{ck0,...,ckr}].


4: Note


This proposition can be used for proving that the barycentric subdivision of any simplicial complex is indeed a simplicial complex.


References


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