From the Steiner Inellipse to the John Ellipsoid of a Simplex: A Corner-Volume Characterization

For a triangle of area $T$, a planar corner-area characterization states that an interior point $M$ lies on the Steiner inellipse precisely when the three corner triangles cut off by the lines through $M$ parallel to the sides have areas $T_1,T_2,T_3$ satisfying $$T_1+T_2+T_3=\frac{1}{2}T\mathrm{.}$$ We give the corresponding statement for a simplex in arbitrary dimension. If $S$ is a nondegenerate $n$-simplex of volume $V$ and $V_1(M),\dots ,V_{n+1}(M)$ are the volumes of the facet-parallel corner simplices determined by $M$, then $$M\in \mathrm{\partial }{E}_J(S)⟺\sum _{i=1}^{n+1}{V}_i(M{)}^{2\mathrm{/}n}=\frac{1}{n}{V}^{2\mathrm{/}n},$$ where $E_J(S)$ is the John ellipsoid of $S$. We also identify the entire corner-volume functional with the central second-moment quadratic of the uniform simplex. The novelty claimed here is limited to the corner-volume formulations and their connections with the planar Steiner-inellipse result; the underlying barycentric, covariance, and John-ellipsoid facts are classical.