Is it true that of all $n$-simplices with edge lengths greater than or equal to some parameter $s$, the regular simplex with edge lengths $s$ has the smallest circumradius? It seems obvious, but I haven't been able to prove this.


I would greatly appreciate if someone could supply a proof or reference. Thank you.



Simplex with edges of length at least s having smallest circumradius
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   $n>1$.  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 0**   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   $\sqrt{\frac{2\cdot n}{n-1}}$).




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Now let's reverse our point of view. Let   $e\_1\ldots e\_k\in H$   be such that   $\|e\_j-e\_k\|\ge s$   for every   $j\ \,k = 1\ldots n$   such that   $j\ne k$,   where   $s>0$   is an arbitrary positive real constant. Furthermore, we may assume that the center of a sphere which has   $e\_1\ldots e\_k$   for its points coincides with the origin of   $H$,   so that   $\|e\_1\|=\ldots =\|e\_n|=r$   for certain positive real   $r$.   Then


$$ s\ \ \le\ \ r\cdot \sqrt{\frac{2\cdot n}{n-1}}$$


i.e.


$$ r\ \ \ge\ \ s\cdot\sqrt{\frac{n-1}{2\cdot n}}$$


The equality holds   $\Leftrightarrow$   all edges have the same length   $\|e\_j-e\_k\|=s$,   and vectors   $e\_1\ldots e\_n$   have to belong to a common $(n-1)$-dimensional linear subspace of   $H$   (substitute *linear* by *affine* in the general case).


