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 Model. Anal. Inform. Sist., 2017, Volume 24, Number 5, Pages 578–595 (Mi mais585)  On $n$-dimensional simplices satisfying inclusions $S\subset [0,1]^n\subset nS$

M. V. Nevskii, A. Yu. Ukhalov

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

Abstract: Let $n\in{\mathbb N}$, $Q_n=[0,1]^n.$ For a nondegenerate simplex $S\subset {\mathbb R}^n$, by $\sigma S$ we denote the homothetic image of $S$ with the center of homothety in the center of gravity of $S$ and ratio of homothety $\sigma$. By $d_i(S)$ we mean the $i$-th axial diameter of $S$, i. e. the maximum length of a line segment in $S$ parallel to the $i$th coordinate axis. Let $\xi(S)=\min \{\sigma\geq 1: Q_n\subset \sigma S\},$ $\xi_n=\min \{ \xi(S): S\subset Q_n \}.$ By $\alpha(S)$ we denote the minimal $\sigma>0$ such that $Q_n$ is contained in a translate of simplex $\sigma S$. Consider $(n+1)\times(n+1)$-matrix $\mathbf{A}$ with the rows containing coordinates of vertices of $S$; the last column of $\mathbf{A}$ consists of 1's. Put $\mathbf{A}^{-1}$ $=(l_{ij})$. Denote by $\lambda_j$ a linear function on ${\mathbb R}^n$ with coefficients from the $j$-th column of $\mathbf{A}^{-1}$, i. e. $\lambda_j(x)= l_{1j}x_1+\ldots+ l_{nj}x_n+l_{n+1,j}.$ Earlier, the first author proved the equalities $\frac{1}{d_i(S)}=\frac{1}{2}\sum_{j=1}^{n+1} |l_{ij}|, \alpha(S) =\sum_{i=1}^n\frac{1}{d_i(S)}.$ In the present paper, we consider the case $S\subset Q_n$. Then all the $d_i(S)\leq 1$, therefore, $n\leq \alpha(S)\leq \xi(S).$ If for some simplex $S^\prime\subset Q_n$ holds $\xi(S^\prime)=n,$ then $\xi_n=n$, $\xi(S^\prime)=\alpha(S^\prime)$, and $d_i(S^\prime)=1$. However, such simplices $S^\prime$ do not exist for all the dimensions $n$. The first value of $n$ with such a property is equal to $2$. For each 2-dimensional simplex, $\xi(S)\geq \xi_2=1+\frac{3\sqrt{5}}{5}=2.34 \ldots>2$. We have an estimate $n\leq \xi_n<n+1$. The equality $\xi_n=n$ takes place if there exists an Hadamard matrix of order $n+1$. Further study showed that $\xi_n=n$ also for some other $n$. In particular, simplices with the condition $S\subset Q_n\subset nS$ were built for any odd $n$ in the interval $1\leq n\leq 11$. In the first part of the paper, we present some new results concerning simplices with such a condition. If $S\subset Q_n\subset nS$, the center of gravity of $S$ coincide, with the center of $Q_n$. We prove that $\sum_{j=1}^{n+1} |l_{ij}|=2 \quad (1\leq i\leq n), \sum_{i=1}^{n} |l_{ij}|=\frac{2n}{n+1} (1\leq j\leq n+1).$ Also we give some corollaries. In the second part of the paper, we consider the following conjecture. Let for simplex $S\subset Q_n$ an equality $\xi(S)=\xi_n$ holds. Then $(n-1)$-dimensional hyperplanes containing the faces of $S$ cut from the cube $Q_n$ the equal-sized parts. Though it is true for $n=2$ and $n=3$, in the general case this conjecture is not valid.

Keywords: $n$-dimensional simplex, $n$-dimensional cube, homothety, axial diameter, interpolation, projection, numerical methods.

 Funding Agency Grant Number The work was supported by the initiative research of Yaroslavl State University VIP-008.

DOI: https://doi.org/10.18255/1818-1015-2017-5-578-595  Full text: PDF file (806 kB) References: PDF file   HTML file

English version:
Automatic Control and Computer Sciences, 2018, 52:7, 667–679 UDC: 514.17+517.51+519.6

Citation: M. V. Nevskii, A. Yu. Ukhalov, “On $n$-dimensional simplices satisfying inclusions $S\subset [0,1]^n\subset nS$”, Model. Anal. Inform. Sist., 24:5 (2017), 578–595; Automatic Control and Computer Sciences, 52:7 (2018), 667–679 Citation in format AMSBIB
\Bibitem{NevUkh17} \by M.~V.~Nevskii, A.~Yu.~Ukhalov \paper On $n$-dimensional simplices satisfying inclusions $S\subset [0,1]^n\subset nS$ \jour Model. Anal. Inform. Sist. \yr 2017 \vol 24 \issue 5 \pages 578--595 \mathnet{http://mi.mathnet.ru/mais585} \crossref{https://doi.org/10.18255/1818-1015-2017-5-578-595} \elib{http://elibrary.ru/item.asp?id=30353169} \transl \jour Automatic Control and Computer Sciences \yr 2018 \vol 52 \issue 7 \pages 667--679 \crossref{https://doi.org/10.3103/S0146411618070192} 

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This publication is cited in the following articles:
1. M. V. Nevskii, A. Yu. Ukhalov, “On optimal interpolation by linear functions on an $n$-dimensional cube”, Automatic Control and Computer Sciences, 52:7 (2018), 828–842    2. M. V. Nevskii, A. Yu. Ukhalov, “Nekotorye svoistva $0/1$-simpleksov”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 18:3 (2018), 305–315   3. M. V. Nevskii, “O nekotorykh zadachakh dlya simpleksa i shara v ${\mathbb R}^n$”, Model. i analiz inform. sistem, 25:6 (2018), 680–691  •  Contact us: math-net2020_02 [at] mi-ras ru Terms of Use Registration Logotypes © Steklov Mathematical Institute RAS, 2020