On a Simplex Inscribed in a Ball

TL;DR

This paper investigates properties of simplices inscribed in a unit ball, focusing on the existence of suitable faces related to the position of their vertices and centers of gravity.

Contribution

It extends previous results by showing that if a vertex of an inscribed simplex is suitable, then suitable faces of all dimensions containing that vertex also exist.

Findings

If some vertex of an inscribed simplex is suitable, then suitable faces of all dimensions containing that vertex exist.

Every simplex inscribed in the unit ball has a suitable face of any dimension if some vertex is suitable.

The paper generalizes earlier results on suitable faces in inscribed 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. Let $S$ be inscribed in $B_n$. If some vertex of $S$ is suitable, then there exists a suitable face of any dimension $\leq n-1$ which contains this vertex.