This article is cited in 4 papers

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 $\|x\|=\left(\sum\limits_{i=1}^n x_i^2 \right)^{1/2}.$ 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 $C=Q_n$.

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

UDC: 514.17+517.51+519.6

Received: 20.09.2018
Revised: 30.10.2018
Accepted: 10.11.2018

DOI: 10.18255/1818-1015-680-691