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   $\sqrt{\frac{2\cdot n}{n-1}}$).