  Let $B$ be a Euclidean ball in ${\mathbb R}^n$ and let $C(B)$ be a space
of~continuous functions $f:B\to{\mathbb R}$ with the uniform norm
$\|f\|_{C(B)}:=\max_{x\in B}|f(x)|.$ By $\Pi_1\left({\mathbb R}^n\right)$ we
mean a set of polynomials of degree $\leq 1$, i.e., a set of linear functions
upon ${\mathbb R}^n$. The interpolation projector $P:C(B)\to \Pi_1({\mathbb
R}^n)$ with the nodes $x^{(j)}\in B$ is defined by the equalities
$Pf\left(x^{(j)}\right)= f\left(x^{(j)}\right)$, $j=1,$ $\ldots,$ $ n+1$. The
norm of $P$ as an operator from $C(B)$ to $C(B)$ can be calculated by the
formula $\|P\|_B=\max_{x\in B}\sum |\lambda_j(x)|.$ Here $\lambda_j$ are the
basic Lagrange polynomials corresponding to the $n$-dimensional nondegenerate
simplex $S$ with the vertices $x^{(j)}$. Let $P^\prime$ be a projector having
the nodes in the vertices \linebreak of a regular simplex inscribed into the
ball. We describe the points $y\in B$ with the property $\|P^\prime\|_B=\sum
|\lambda_j(y)|$. Also we formulate a geometric conjecture which implies that
$\|P^\prime\|_B$ is equal to the minimal norm of an interpolation projector
with nodes in $B$. We prove that this conjecture holds true at least for
$n=1,2,3,4$.
  Keywords: regular simplex, ball, linear interpolation, projector, norm
