Geometric estimates in interpolation on an $n$-dimensional ball

Suppose $n\in {\mathbb N}$. Let $B_n$ be a Euclidean unit ball in ${\mathbb R}^n$ given by the inequality $\|x\|\leq 1$, $\|x\|:=\left(\sum\limits_{i=1}^n x_i^2\right)^{\frac{1}{2}}$. By $C(B_n)$ we mean a set of continuous functions $f:B_n\to{\mathbb R}$ with the norm $\|f\|_{C(B_n)}:=\max\limits_{x\in B_n}|f(x)|$. The symbol $\Pi_1\left({\mathbb R}^n\right)$ denotes a set of polynomials in $n$ variables of degree $\leq 1$, i. e., linear functions upon ${\mathbb R}^n$. Assume that $x^{(1)}, \ldots, x^{(n+1)}$ are vertices of an $n$-dimensional nondegenerate simplex $S\subset B_n$. The interpolation projector $P:C(B_n)\to \Pi_1({\mathbb R}^n)$ corresponding to $S$ is defined by the equalities $Pf\left(x^{(j)}\right)=
f\left(x^{(j)}\right).$ Denote by $\|P\|_{B_n}$ the norm of $P$ as an operator from $C(B_n)$ onto $C(B_n)$. Let us define $\theta_n(B_n)$ as the minimal value of $\|P\|_{B_n}$ under the condition $x^{(j)}\in B_n$. We describe the approach in which the norm of the projector can be estimated from the bottom through the volume of the simplex. Let $\chi_n(t):=\frac{1}{2^nn!}\left[ (t^2-1)^n \right] ^{(n)}$ be the standardized Legendre polynomial of degree $n$. We prove that $ \|P\|_{B_n} \geq \chi_n^{-1} \left(\frac{\mathrm{vol}(B_n)}{\mathrm{vol}(S)}\right).$ From this, we obtain the equivalence $\theta_n(B_n)$ $\asymp$ $\sqrt{n}$. Also we estimate the constants from such inequalities and give the comparison with the similar relations for linear interpolation upon the $n$-dimensional unit cube. These results have applications in polynomial interpolation and computational geometry.