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Definitions

Let's consider a full $n$-dimensional simplicial complex $C$ in $\ell_2(X)$. By that I mean a set of all functions $f:X \to[0,1]$ such that $\sum_{x\in X}f(x)=1$ and there are at most $n+1$ points $x$ such that $f(x)\neq 0$. Functions $f$ such that $f(x) = 1$ for some $x$ are called vertices. A simplex is a convex hull of some vertices - $m$-simplex is a convex hull of $m+1$ different vertices.

Barycentric subdivision of $C$ is a family $\beta C$ of its subsets defined below. $$\beta C=\{\mathrm{conv}(v_0, \mathrm{av}(v_0,v_1), \mathrm{av}(v_0, v_1, v_2), \ldots, \mathrm{av}(v_0,\ldots, v_n) \ |\ v_i - \mathrm{vertex} \},$$ where conv is a convex hull and av is the arithmetic mean. Note that if we think of functions $\mathrm{av}(v_0,\ldots,v_k)$ as vertices, then $s\in \beta C$ is a $n$-simplex with vertices of that form.

We get second barycentric subdivision $\beta^2 C$, if we divide each $s\in \beta C$ into pieces using the procedure shown above and take all those pieces.

A star around a point $f\in C$ from a set of simplices $S \subseteq C$ is $$St(f,S)=\bigcup \{s \in S\ |\ f \in s\}.$$

Two subsets of $C$ are $c$-distant if their distance $dist$ is no less than $c$. $dist(A,B)=\inf \Big\{\sqrt{\sum_{x \in X} (a(x)-b(x))^2} \ | \ a\in A, b\in B \Big\}$

Problem

In Asymptotic dimension in Będlewo (top of the 5th page) it is said that there exists a constant $c$ such that $St(av(v_0,\ldots,v_k),\beta^2 C)$ is $c$-distant from $St(av(v'_0,\ldots,v'_k), \beta^2 C)$ provided $(v_0,\ldots,v_k)\neq(v'_0,\ldots,v'_k)$. I was able to show that any simplex $s\in \beta C$ such that $(v_0,\ldots,v_k) \in h$ may intersect with $s'$ such that $(v'_0,\ldots,v'_k) \in s'$ only within $\mathrm{conv}((w_i)_{i=1}^l)$, where $w_i = v_p = v'_q$ for some $p,q$. Using that, I show that $St(av(v_0,\ldots,v_k),\beta^2 C) \cap s \cap \mathrm{conv}((w_i)_{i=1}^l)$ is empty, so there exists $\varepsilon$, such that $St(av(v_0,\ldots,v_k),\beta^2 C) \cap s$ and $St(av(v'_0,\ldots,v'_k),\beta^2 C) \cap s'$ are $\varepsilon$-distant. Finally I notice that there are finitely many different 'cases', so the infimum of all different $\varepsilon$s is positive and that may be used as $c$.

I'm not happy with my abstract proof and would really appreciate one with exact bounds for $c$, but I have no good idea.

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