NDLI: On one problem for a simplex and a cube in ℝ$^{ n }$

Content Provider	Springer Nature Link
Author	Nevskii, M. V.
Copyright Year	2014
Abstract	Assume that S is a nondegenerate simplex in ℝ$^{ n }$. Denote by α(S) the minimum σ > 0 such that the unit cube Q $_{ n }$:= [0, 1]$^{ n }$ belongs to a translate of σS. In the case of σ(S) ≠ 1, a translate of α(S)S containing Q $_{ n }$ is the image of S under homothety with the center at some point x ∈ ℝ$^{ n }$. In this paper, we present the following computational formula for x. Denote by x $^{(j)}$ (j = 1, ..., n + 1) the vertices of S. Let A be an n + 1-by-n + 1 matrix such that its rows contain the coordinates of x $^{(j)}$; here, the last column of A consists of ones. Assume that A $^{−1}$ = (l $_{ ij }$). Then, the coordinates of x are the numbers $$x_k = \frac{{\sum\nolimits_{j = 1}^{n + 1} {\left( {\sum\nolimits_{i = 1}^n {\left\| {l_{ij} } \right\|} } \right)\left( {x_k^{(j)} - 1} \right)} }} {{\sum\nolimits_{i = 1}^n {\sum\nolimits_{j = 1}^{n + 1} {\left\| {l_{ij} } \right\| - 2} } }} (k = 1, \ldots ,n).$$ Since α(S) ≠ 1, the denominator in the right-hand side of this equality is distinct from zero. In addition, we present estimates for norms of projections under the linear interpolation of continuous functions given on Q $_{ n }$.
Starting Page	521
Ending Page	527
Page Count	7
File Format	PDF
ISSN	01464116
Journal	Automatic Control and Computer Sciences
Volume Number	48
Issue Number	7
e-ISSN	1558108X
Language	English
Publisher	Allerton Press
Publisher Date	2015-02-01
Publisher Place	Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	n-dimensional simplex n-dimensional cube axial diameter homothety interpolation projector Control Structures and Microprogramming
Content Type	Text
Resource Type	Article
Subject	Signal Processing Control and Systems Engineering Software

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