On the minimal positive homothetic image of a simplex containing a convex body

Невский, М. В.

doi:10.1134/s0001434613030140

Cited by 4 publications

(3 citation statements)

References 5 publications

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“…The Center of this segment is the point (1.5) In [7], it is found that A certain generalization of (1.6) is given in [5]. The present paper continues this cycle of my works.…”

Section: Introductionsupporting

confidence: 66%

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On one problem for a simplex and a cube in ℝ n

Невский

2014

Aut. Control Comp. Sci.

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“…The Center of this segment is the point (1.5) In [7], it is found that A certain generalization of (1.6) is given in [5]. The present paper continues this cycle of my works.…”

Section: Introductionsupporting

confidence: 66%

“…Since the minimum positive homothetic image of S that contains Q n is a translate of σS, then Q is the unique maximum homothetic image of Q n that belongs to S. (Various ways of illustrating this fact are pre sented in [5].) The translate of σS that contains Q n coincides with S x,σ .…”

Section: Determining the Center Of Homothetymentioning

confidence: 99%

On one problem for a simplex and a cube in ℝ n

Невский

2014

Aut. Control Comp. Sci.

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“…The basic theoretical apparatus in this area was developed by M. Nevsky approximately in 2000-2014. Here we note the monograph [26] containing many results and links, the papers [14]- [15], [17]- [23], [25], [27]- [29], and also the dissertation [30]. A. Ukhalov joined the research starting around 2015.…”

Section: Introductionmentioning

confidence: 99%

Estimates for Interpolation Projectors and Related Problems in Computational Geometry

Nevskii,

Ukhalov

2021

Preprint

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This paper contains a survey of results obtained by the authors mostly during the past few years and published by 2021. In particular, we present the best of known estimates of numerical characteristics related to the research theme.Sections: 1. Introduction. 2. The case when n + 1 is an Hadamard number. 3. Estimates for the minimal absorption index of a cube by a simplex. 4. Estimates for the minimal norm of a projector in linear interpolation on a cube in R n . 5. Estimates of numbers ξ n and θ n . 6. Simplices satisfying the inclusions S ⊂ Q n ⊂ nS. 7. Perfect simplices. 8. Equisecting simplices. 9. Properties of (0, 1)-matrices of order n having maximal determinant. 10. Problems for a simplex and a Euclidean ball. 11. Linear interpolation on a Euclidean ball.Bibliography: 56 titles.

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On Some Problems for a Simplex and a Ball in R<sup>n</sup>

Невский

2018

Model. anal. inf. sist.

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Let \(C\) be a convex body and let \(S\) be a nondegenerate simplex in \({\mathbb R}^n\). Denote by \(\tau S\) the image of \(S\) under homothety with a center of homothety in the center of gravity of \(S\) and the ratio \(\tau\). We mean by \(\xi(C;S)\) the minimal \(\tau>0\) such that \(C\) is a subset of the simplex \(\tau S\). Define \(\alpha(C;S)\) as the minimal \(\tau>0\) such that \(C\) is contained in a translate of \(\tau S\). Earlier the author has proved the equalities \(\xi(C;S)=(n+1)\max\limits_{1\leq j\leq n+1}\max\limits_{x\in C}(-\lambda_j(x))+1\) (if \(C\not\subset S\)), \(\alpha(C;S)=\sum\limits_{j=1}^{n+1} \max\limits_{x\in C} (-\lambda_j(x))+1.\)Here \(\lambda_j\) are the linear functions that are called the basic Lagrange polynomials corresponding to \(S\). The numbers \(\lambda_j(x),\ldots, \lambda_{n+1}(x)\) are the barycentric coordinates of a point \(x\in{\mathbb R}^n\). In his previous papers, the author investigated these formulae in the case when \(C\) is the \(n\)-dimensional unit cube \(Q_n=[0,1]^n\). The present paper is related to the case when \(C\) coincides with the unit Euclidean ball \(B_n=\{x: \|x\|\leq 1\},\) where \(\|x\|=\left(\sum\limits_{i=1}^n x_i^2 \right)^{1/2}.\) We establish various relations for \(\xi(B_n;S)\) and \(\alpha(B_n;S)\), as well as we give their geometric interpretation. For example, if \(\lambda_j(x)=l_{1j}x_1+\ldots+l_{nj}x_n+l_{n+1,j},\) then \(\alpha(B_n;S)=\sum\limits_{j=1}^{n+1}\left(\sum\limits_{i=1}^n l_{ij}^2\right)^{1/2}\). The minimal possible value of each characteristics \(\xi(B_n;S)\) and \(\alpha(B_n;S)\) for \(S\subset B_n\) is equal to \(n\). This value corresponds to a regular simplex inscribed into \(B_n\). Also we compare our results with those obtained in the case \(C=Q_n\).

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On the minimal positive homothetic image of a simplex containing a convex body

Cited by 4 publications

References 5 publications

On one problem for a simplex and a cube in ℝ n

On one problem for a simplex and a cube in ℝ n

Estimates for Interpolation Projectors and Related Problems in Computational Geometry

On Some Problems for a Simplex and a Ball in R<sup>n</sup>

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