RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Model. Anal. Inform. Sist.: Year: Volume: Issue: Page: Find

 Model. Anal. Inform. Sist., 2018, Volume 25, Number 6, Pages 680–691 (Mi mais656)

Computational Geometry

On some problems for a simplex and a ball in ${\mathbb R}^n$

M. V. Nevskii

P.G. Demidov Yaroslavl State University, 14 Sovetskaya str., Yaroslavl 150003, Russia

Abstract: 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 $\|x\|=(\sum\limits_{i=1}^n x_i^2 )^{1/2}.$ 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 $\alpha(B_n;S)= \sum\limits_{j=1}^{n+1}(\sum\limits_{i=1}^n l_{ij}^2)^{1/2}$. 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 $C=Q_n$.

Keywords: $n$-dimensional simplex, $n$-dimensional ball, homothety, absorption index.

DOI: https://doi.org/10.18255/1818-1015-680-691

Full text: PDF file (571 kB)
References: PDF file   HTML file

UDC: 514.17+517.51+519.6
Revised: 30.10.2018
Accepted:10.11.2018

Citation: M. V. Nevskii, “On some problems for a simplex and a ball in ${\mathbb R}^n$”, Model. Anal. Inform. Sist., 25:6 (2018), 680–691

Citation in format AMSBIB
\Bibitem{Nev18} \by M.~V.~Nevskii \paper On some problems for a simplex and a ball in ${\mathbb R}^n$ \jour Model. Anal. Inform. Sist. \yr 2018 \vol 25 \issue 6 \pages 680--691 \mathnet{http://mi.mathnet.ru/mais656} \crossref{https://doi.org/10.18255/1818-1015-680-691} 

• http://mi.mathnet.ru/eng/mais656
• http://mi.mathnet.ru/eng/mais/v25/i6/p680

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. M. V. Nevskii, A. Yu. Ukhalov, “Lineinaya interpolyatsiya na evklidovom share v ${\mathbb R}^n$”, Model. i analiz inform. sistem, 26:2 (2019), 279–296
•  Number of views: This page: 66 Full text: 32 References: 12