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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2505.15739 |
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Sommario:
- 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.