Geodesic
On Some Problem for a Simplex and a Cube in ${\mathbb R}^n$
Modelirovanie i analiz informacionnyh sistem, Tome 20 (2013) no. 3, pp. 77-85.

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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 $$x_k= \frac{\sum_{j=1}^{n+1} \left(\sum_{i=1}^n \left|l_{ij}\right|\right)x^{(j)}_k -1} {\sum_{i=1}^n\sum_{j=1}^{n+1} |l_{ij}|- 2} \quad (k=1,\ldots,n).$$ 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, projection.
Mots-clés : interpolation 
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M. V. Nevskii. On Some Problem for a Simplex and a Cube in ${\mathbb R}^n$. Modelirovanie i analiz informacionnyh sistem, Tome 20 (2013) no. 3, pp. 77-85. http://geodesic.mathdoc.fr/item/MAIS_2013_20_3_a4/

[1] M. V. Nevskij, “Minimal projections and largest simplices”, Modeling and Analysis of Information Systems, 14:1 (2007), 3–10 (in Russian)

[2] M. V. Nevskii, “On a property of $n$-dimensional simplices”, Math. Notes, 87:4 (2010), 543–555 | DOI | DOI | MR | Zbl

[3] M. V. Nevskii, “On the axial diameters of a convex body”, Math. Notes, 90:2 (2011), 295–298 | DOI | DOI | MR | Zbl

[4] M. V. Nevskii, Geometricheskie ocenki v polinomialnoi interpolyacii, YarGU, Yaroslavl, 2012, 218 pp. (in Russian)

[5] M. V. Nevskii, “On the minimal positive homothetic image of a simplex containing a convex body”, Math. Notes, 93:3 (2013), 112–120 | DOI | MR | Zbl

[6] M. Hudelson, V. Klee, D. Larman, “Largest $j$-simplices in $d$-cubes: some relatives of the Hadamard maximum determinant problem”, Linear Algebra Appl., 241–243 (1996), 519–598 | DOI | MR | Zbl

[7] M. Nevskii, “Properties of axial diameters of a simplex”, Discrete Comput. Geom., 46:2 (2011), 301–312 | DOI | MR | Zbl

[8] P. R. Scott, “Lattices and convex sets in space”, Quart. J. Math. Oxford (2), 36 (1985), 359–362 | DOI | MR | Zbl

[9] P. R. Scott, “Properties of axial diameters”, Bull. Austral. Math. Soc., 39 (1989), 329–333 | DOI | MR | Zbl