On Some Problems for a Simplex and a Ball in Rn

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 \(C=Q_n\).