Will the upper bound $b=2$ work in a simplicial proof of Brouwer's Fixed Point Theorem, Hurewicz' and Wallman's "Dimension Theory"?

Question

Background:

I am self studying Hurewicz' and Wallman's 'Dimension Theory'. My question concerns pp. 37-39 there within. The theorem A) below is equivalent to Brouwer's FPT, hence the title. I will type verbatim some relevant portions of the text to ease the reader's burden.

Theorem Statement:

A) Let $S_n$ be an $n$-sphere, e.g., the set of all points in $E_{n+1}$ at distance 1 from the origin. Then there exists no function $f_t(x)$ of the two variables $x\in S_n$ and $0\leq t\leq 1$, with values in $S_n$, continuous in the pair $(x,t)$, and with the boundary conditions \begin{equation} f_0(x)=x\qquad\textrm{and}\qquad f_1(x)=\textrm{constant}.^\dagger \end{equation} $\dagger$: $f_1$ is a constant mapping; in general, a mapping $f$ of a space $X$ in a space $Y$ is called a $\textit{constant}$ mapping if $f$ sends all of $X$ into a single point of $Y$.

(Beginning of) Proof:

We use the simple geometrical fact that if $p_0,\dots,p_n$ are $n+1$ points on the unit sphere $S_n$ whose distances in pairs are less than 1, then $p_0,\dots,p_n$ determine a unique (perhaps degenerate) spherical $n$-simplex.

My question:

Why choose $1=b$ instead of the more obvious upper bound of $2=b$ in order to ensure that any subset of $S_n$ with $n+1$ members, and with set diameter less than $b$ determines a unique spherical $n$-simplex on $S_n$?

Answer 1

Because there is no natural way to determine a unique spherical $n$-simplex if $b=2$ once $n\ge2$.

Here is an example.

Let $n=2$. Consider points $p_0=(1,0,0)$, $p_1=(-\frac12, \frac{\sqrt3}2,0)$, $p_2=(-\frac12, -\frac{\sqrt3}2,0)$ that are evenly distributed on equator $\mathcal C$, the big circle of $S_2$ that passes through them. There are $3$ minimal spherical $2$-simplexes on $S_2$ that contains $p_0,p_1,p_2$:

• the major arc from $p_0$ to $p_2$ on $\mathcal C$.
• the major arc from $p_1$ to $p_0$ on $\mathcal C$.
• the major arc from $p_2$ to $p_1$ on $\mathcal C$.

There is no natural way to pick one of the $3$ (degenerate) spherical $2$-simplexes as the one that is determined by $p_0,p1,p_2$.

In case you would prefer an example of non-degenerate spherical simplexes, just move $p_0,p_1,p_2$ a little towards the north pole.