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
$C=Q_n$.
