On a Simplex Contained in a Ball

TL;DR

The paper studies the mutual placement of the unit ball $B_n$ and the minimal-volume ellipsoid $E$ containing a nondegenerate simplex $S\subset B_n$, focusing on the centers of gravity of faces and their opposite faces to define points $y_J$ on $\partial E$. It refines a prior result by proving that if some vertex of $S$ is suitable (i.e., yields $y_{\{j\}}\in B_n$), then for every $m\in\{1,\dots, n-1\}$ there exists a suitable $m$-dimensional face containing that vertex. The proof leverages affine-invariance to reduce to the inscribed-ball case and employs a careful averaging argument over all $J'={1,j_1,\dots,j_m}$ to show the existence of a face with $y_{J'}\in B_n$, with a key ratio $b/a=(n-1)/2$ arising in the analysis. These results connect to polynomial interpolation and optimal projector norms, and provide insight into configurations where the minimal ellipsoid coincides with the ball, notably for regular simplices.

Abstract

Let $B_n$ be the $n$-dimensional unit ball given by the inequality $\|x\|\leq 1$, where $\|x\|$ is the standard Euclid norm in ${\mathbb R}^n$. For an $n$-dimensional nondegenerate simplex $S$, we denote by $E$ the ellipsoid of minimum volume which contains $S$. Suppose $S\subset B_n$, $0\leq m\leq n-1$. Let $G$ be any $m$-dimensional face of $S$ and let $H$ be the opposite $(n-m-1)$-dimensional face. Denote by $g$ and $h$ the centers of gravity of $G$ and $H$ respectively. Define $y$ as the intersection point of the line passing from $g$ to $h$ with the boundary of $E$. Let us call the face $G$ suitable if $y\in B_n.$ Earlier it was proved that each simplex $S\subset B_n$ has a suitable face of any dimension $\leq n-1$. We show the following. If some vertex of $S$ is suitable, then there exists a suitable face of any dimension $\leq n-1$ which contains this vertex.