I am currently reading a paper on topological data analysis which includes som discrete Morse theory arguments. I got stuck on a corollary that in the paper I'm reading is described as simply following from a lemma (*).
Denote by $P$ an almost linear metric space with distinguished point p. We say that a nonempty finite metric space $P$ is almost linear if there is a point $p \in P$ (distinguished point) such that $P\setminus \{p\}$ is isometric to a finite subset of $\mathbb{R}$. We consider the discrete gradient vector field $X$, defined as follows on the Vietoris-Rips complex $\mathcal{V}(P)_r$. We use the convention that $p$ is the smallest point and for a simplex $[a_1,\ldots,a_j]$: $a_1<\ldots<a_j$.
Definition of $X$: If a simplex $\sigma = [a_1,a_2,\ldots,a_j]$ in $\mathcal{V}(P)_r$ has a coface $a_0 \cup \sigma:= [a_0,a_1,a_2,\ldots,a_j]$ with $a_0 < a_1$, then $X$ matches $\sigma$ to $a_0 \cup \sigma$ with $a_0$ as small as possible. $X$ matches no other simplices.
(*)One can show that for $j \geq 2$, a simplex $[a_1,\ldots,a_j]$ is critical in $X$ if and only if the following conditions are satisfied:
- $a_1 \neq p$,
- $[q,a_1,\ldots,a_j] \notin \mathcal{V}(P)_r$ for any $q < a_1$,
- $[p,a_2,a_3,\ldots,a_j] \in \mathcal{V}(P)_r$.
Now I want to show, using the above, that if $[a_1,a_2,a_3,\ldots,a_j]$ is a critical simplex in $X$, and under the assumption that there does not exist a vertex $b$ such that $[p,b] \in \mathcal{V}(P)_r$ and $a_1 < b < a_2$, then the simplex $[a_1,a_3,\ldots,a_j]$ is critical in $X$.
The first item follows directly by definition and so does the third since $[p,a_3,\ldots,a_j]$ is a subsimplex of $[p,a_2,a_3,\ldots,a_j]$. However, I struggle with showing that the second item is satisfied, as well.
I want to perform a contradiction argument. Assume that $[a_1,\ldots,a_j]$ is critical and $[q,a_1,a_3,\ldots,a_j] \in \mathcal{V}(P)_r$ for some $q<a_1$. Then $q \neq p$ since $[a_1,\ldots,a_j]$ is critical, so $[p,a_1] \notin \mathcal{V}(P)_r$. I see that we can "use" $b$ to see that when adding $a_2$ to the simplex, the distance between $q$ and $a_2$ is precisely the Euclidean distance. However, I do not find a way on how to proceed in order to obtain the desired result.
I really appreciate any hint on how I could prove that the second item is satisfied by $[a_1,a_3,\ldots,a_j]$!
