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 Model. Anal. Inform. Sist., 2013, Volume 20, Number 3, Pages 77–85 (Mi mais312)

On Some Problem for a Simplex and a Cube in ${\mathbb R}^n$

M. V. Nevskii

P. G. Demidov Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, Russia

Abstract: Let $S$ be a nondegenerate simplex in ${\mathbb R}^n$. Denote by $\alpha(S)$ the minimal $\sigma>0$ such that the unit cube $Q_n:=[0,1]^n$ is contained in a translate of $\sigma S$. In the case $\alpha(S)\ne 1$ the translate of $\alpha(S)S$ containing $Q_n$ is a homothetic copy of $S$ with the homothety center at some point $x\in{\mathbb R}^n$. We obtain the following computational formula for $x$. Denote by $x^{(j)}$ $(j=1,\ldots, n+1)$ the vertices of $S$. Let ${\mathbf A}$ be the matrix of order $n+1$ with the rows consisting of the coordinates of $x^{(j)};$ the last column of ${\mathbf A}$ consists of 1's. Suppose that ${\mathbf A}^{-1}=(l_{ij}).$ Then the coordinates of $x$ are the numbers
$$x_k= \frac{\sum_{j=1}^{n+1} (\sum_{i=1}^n |l_{ij}|)x^{(j)}_k -1} {\sum_{i=1}^n\sum_{j=1}^{n+1} |l_{ij}|- 2} \quad (k=1,\ldots,n).$$
Since $\alpha(S)\ne 1,$ the denominator from the right-hand part of this equality is not equal to zero. Also we give the estimates for norms of projections dealing with the linear interpolation of continuous functions defined on $Q_n$.

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

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UDC: 514.17+517.51

Citation: M. V. Nevskii, “On Some Problem for a Simplex and a Cube in ${\mathbb R}^n$”, Model. Anal. Inform. Sist., 20:3 (2013), 77–85

Citation in format AMSBIB
\Bibitem{Nev13} \by M.~V.~Nevskii \paper On Some Problem for a Simplex and a Cube in ${\mathbb R}^n$ \jour Model. Anal. Inform. Sist. \yr 2013 \vol 20 \issue 3 \pages 77--85 \mathnet{http://mi.mathnet.ru/mais312}