On Some Problems for a Simplex and a Ball in R<sup>n</sup>
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 \(C\)...
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| Format: | Article |
| Language: | English |
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Yaroslavl State University
2018-12-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/766 |
| _version_ | 1797878012164702208 |
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| author | Mikhail V. Nevskii |
| author_facet | Mikhail V. Nevskii |
| author_sort | Mikhail V. Nevskii |
| collection | DOAJ |
| description | 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 \(C=Q_n\). |
| first_indexed | 2024-04-10T02:26:00Z |
| format | Article |
| id | doaj.art-59d7df89f7a64f3e8a9084448610ac87 |
| institution | Directory Open Access Journal |
| issn | 1818-1015 2313-5417 |
| language | English |
| last_indexed | 2024-04-10T02:26:00Z |
| publishDate | 2018-12-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj.art-59d7df89f7a64f3e8a9084448610ac872023-03-13T08:07:29ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172018-12-0125668069110.18255/1818-1015-2018-6-680-691534On Some Problems for a Simplex and a Ball in R<sup>n</sup>Mikhail V. Nevskii0Ярославский государственный университет им. П.Г. ДемидоваLet \(C\) 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 \(C=Q_n\).https://www.mais-journal.ru/jour/article/view/766n-мерный симплексn-мерный шаргомотетиякоэффициент поглощения |
| spellingShingle | Mikhail V. Nevskii On Some Problems for a Simplex and a Ball in R<sup>n</sup> Моделирование и анализ информационных систем n-мерный симплекс n-мерный шар гомотетия коэффициент поглощения |
| title | On Some Problems for a Simplex and a Ball in R<sup>n</sup> |
| title_full | On Some Problems for a Simplex and a Ball in R<sup>n</sup> |
| title_fullStr | On Some Problems for a Simplex and a Ball in R<sup>n</sup> |
| title_full_unstemmed | On Some Problems for a Simplex and a Ball in R<sup>n</sup> |
| title_short | On Some Problems for a Simplex and a Ball in R<sup>n</sup> |
| title_sort | on some problems for a simplex and a ball in r sup n sup |
| topic | n-мерный симплекс n-мерный шар гомотетия коэффициент поглощения |
| url | https://www.mais-journal.ru/jour/article/view/766 |
| work_keys_str_mv | AT mikhailvnevskii onsomeproblemsforasimplexandaballinrsupnsup |