Let $H$ be a Hilbert space. Let $e_1\ \ldots\ e_n$ be vectors such that $\forall_{k=1\ldots n}\ \|e_k\|\le 1$ for a natural $n$ (to me natural numbers are always positive). I am addressing $(n-1)$-simplex for the sake of avoiding eyesores. Then:
$$ 0\ \le\ (e_1+\ldots +e_n)^2\ \ \le\ \ n\ +\ 2\cdot\sum_{j\ne k}e_j\cdot e_k $$
It follows that there exist $j\ne k$ such that
$$ e_j\cdot e_k\ \ \ge\ \ \frac{-n}{2\cdot\binom n2}\ =\ \frac1{1-n}$$
and
$$\|e_j-e_k\|\ =\ \sqrt{e_j^2 - 2\cdot e_j\cdot e_k + e_k^2}\ \le\ \sqrt{2\cdot(1+\frac 1{n-1})}\ =\ \sqrt{\frac{2\cdot n}{n-1}}$$
i.e.
$$\|e_j-e_k\|\ \le\ \sqrt{\frac{2\cdot n}{n-1}}$$
REMARK We get equality above $\Leftrightarrow$ $\|e_j\|=\|e_k\|=1$ and the $\binom n2$ values $e_j\cdot e_k$ are all equal one to another, i.e. all distances $\|e_j-e_k\|$ are all equal. (Then of course all dot products are equal to $\frac 1{1-n}$, and the distances to $\sqrt{\frac{2\cdot n}{n-1}}$).