On Some Problems Related to a Simplex and a Ball

Let $C$ be a convex body and let $S$ be a nondegenerate simplex in ${\mathbb R}^n$\mathrm{.}Denoteby$\xi(C;S)$theminimal$\tau>0$suchthat$C$ is a subset of the simplex $\tau S$. By $\alpha (C;S)$ we mean 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 linear functions called the basic Lagrange polynomials corresponding to $S$. In his previous papers, the author has investigated these formulae if $C=[0,1{]}^n$. The present paper is related to the case when $C$ coincides with the unit Euclidean ball $B_n=\{x:\mathrm{\parallel }x\mathrm{\parallel }\le 1\},$ where $\mathrm{\parallel }x\mathrm{\parallel }={\left({\sum }_{i=1}^n{x}_i^2\right)}^{1\mathrm{/}2}\mathrm{.}$ We establish various relations for $\xi (B_n;S)$ and $\alpha (B_n;S)$, as well as we give their geometric interpretation.