On Some Problem for a Simplex and a Cube in Rⁿ
https://doi.org/10.18255/1818-1015-2013-3-77-85
Abstract
Let S be a nondegenerate simplex in Rⁿ. Denote by α(S) the minimal σ > 0 such that the unit cube Qn:= [0, 1]ⁿ is contained in a translate of σS. In the case α(S) ≠ 1 the translate of α(S)S containing Qn is a homothetic copy of S with the homothety center at some point x ∈ Rⁿ . We obtain the following computational formula for x. Denote by x (j) (j = 1, . . . , n+ 1) the vertices of S. Let A be the matrix of order n+ 1 with the rows consisting of the coordinates of x (j) ; the last column of A consists of 1’s. Suppose that A−1 = (lIj ). Then the coordinates of x are the numbers
xk = Pn+1 j=1 ( Pn i=1 |lij |) x (j) k − 1 Pn i=1 Pn+1 j=1 |lij | − 2 (k = 1, . . . , n).
Since α(S) ≠ 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 Qn.
About the Author
M. V. Nevskii
P.G. Demidov Yaroslavl State University
Russian Federation
Russian Federation
канд. физ.-мат. наук, доцент, декан математического факультета,
Sovetskaya str., 14, Yaroslavl, 150000, Russia
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For citations:
Nevskii M.V. On Some Problem for a Simplex and a Cube in Rⁿ. Modeling and Analysis of Information Systems. 2013;20(3):77-85. (In Russ.) https://doi.org/10.18255/1818-1015-2013-3-77-85
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