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 Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform.: Year: Volume: Issue: Page: Find

 Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2018, Volume 18, Issue 3, Pages 305–315 (Mi isu765)

Scientific Part
Mathematics

Some properties of $0/1$-simplices

M. V. Nevskii, A. Yu. Ukhalov

P. G. Demidov Yaroslavl State University, 14, Sovetskaya Str., Yaroslavl, 150003, Russia

Abstract: Let $n\in {\mathbb N}$, and let $Q_n=[0,1]^n$. For a nondegenerate simplex $S\subset {\mathbb R}^n$, by $\sigma S$ we mean the homothetic copy of $S$ with center of homothety in the center of gravity of $S$ and ratio of homothety $\sigma$. Put $\xi(S)=\min\{\sigma\geq 1: Q_n\subset \sigma S\}$, $\xi_n=\min\{\xi(S): S\subset Q_n\}$. By $P$ we denote the interpolation projector from $C(Q_n)$ onto the space of linear functions of $n$ variables with the nodes in the vertices of a simplex $S\subset Q_n$. Let $\|P\|$ be the norm of $P$ as an operator from $C(Q_n)$ to $C(Q_n)$, $\theta_n=\min\|P\|$. By $\xi^\prime_n$ and $\theta^\prime_n$ we denote the values analogous to $\xi_n$ and $\theta_n$, with the additional condition that corresponding simplices are $0/1$-polytopes, i. e., their vertices coincide with vertices of $Q_n$. In the present paper, we systematize general estimates of the numbers $\xi^\prime_n$, $\theta^\prime_n$ and also give their new estimates and precise values for some $n$. We prove that $\xi^\prime_n\asymp n$, $\theta^\prime_n\asymp \sqrt{n}$. Let one vertex of $0/1$-simplex $S^*$ be an arbitrary vertex $v$ of $Q_n$ and the other $n$ vertices are close to the vertex of the cube opposite to $v$. For $2\leq n\leq 5$, each simplex extremal in the sense of $\xi^\prime_n$ coincides with $S^*$. The minimal $n$ such that $\xi(S^*)>\xi^\prime_n$ is equal to $6$. Denote by $P^*$ the interpolation projector with the nodes in the vertices of $S^*$. The minimal $n$ such that $\|P^*\|>\theta^\prime_n$ is equal to $5$.

Key words: simplex, cube, homothety, axial diameter, interpolation, projector, numerical methods.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 1.12873.2018/12.1 This work was carried out within the framework of the State Programme of the Ministry of Education and Science of the Russian Federation (project no. 1.12873.2018/12.1).

DOI: https://doi.org/10.18500/1816-9791-2018-18-3-305-315

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UDC: 514.17+517.51+519.6

Citation: M. V. Nevskii, A. Yu. Ukhalov, “Some properties of $0/1$-simplices”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 18:3 (2018), 305–315

Citation in format AMSBIB
\Bibitem{NevUkh18} \by M.~V.~Nevskii, A.~Yu.~Ukhalov \paper Some properties of $0/1$-simplices \jour Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform. \yr 2018 \vol 18 \issue 3 \pages 305--315 \mathnet{http://mi.mathnet.ru/isu765} \crossref{https://doi.org/10.18500/1816-9791-2018-18-3-305-315} \elib{http://elibrary.ru/item.asp?id=35728998}