Ratio of inscribed/circumscribed ellipsoids: geometrical proof?

Question

Let $K$ be a convex subset of ${\mathbb R}^n$, with non-void interior. The Löwner-John theorem states that there are a minimal volume ellipsoid $\cal E$ containing $K$, a maximal one $\cal F$ contained in $K$, and that $\frac1n\cal E\subset\cal F$. The latter property is sharp since it is an equality when $K$ is a simplex.

The simplex case reduces to the regular one, for which $\cal E$ and $\cal F$ are concentric spheres of radii $R$ and $r$. Actually, this implies the inclusion part of Löwner-John theorem for general convex bodies. The fact that $R=nr$ follows from a cumbersome calculation.

Is there a calculus-free proof of $R=nr$ in the simplex case ? Presumably a purely geometrical proof.

Answer 1

Let $S$ be a regular simplex with vertices $x_0,\dots,x_n$ and center $c$, and let $S’$ be the simplex with vertices $c,x_1,\dots,x_n$. Let $c’$ be the center of their shared facet $x_1,\dots,x_n$. Since $S$ can be decomposed into $n+1$ simplices congruent to $S’$, we have $\DeclareMathOperator{\Vol}{Vol} \Vol S = (n+1)\Vol S’$, which implies $|x_0-c’|=(n+1)|c-c’|$, or equivalently $|x_0-c|=n|c-c’|$. Now observe that $|x_0-c|$ is the circumradius and $|c-c’|$ is the inradius.