Geometric Estimates in Interpolation by Linear Functions on a Euclidean Ball

Let $B_n$ be the Euclidean unit ball in ${\mathbb{R}}^n$ given by the inequality $\mathrm{\parallel }x\mathrm{\parallel }\le 1$, $|x|:=\left(\sum\limits_{i=1}^n x_i^2\right)^{\frac{1}{2}}$\mathrm{.}By$C(B_n)$ we mean the space of continuous functions $f:B_n\to \mathbb{R}$ with the norm $|f|{C(B_n)} := \max\limits{x\in B_n}|f(x)|$\mathrm{.}Thesymbol$\Pi_1\left({\mathbb R}^n\right)$denotesthesetofpolynomialsin$n$variablesofdegree$\leq 1$, i.e., the set of linear functions upon ${\mathbb{R}}^n$. Assume $x^{(1)}, \ldots, x^{(n+1)}$aretheverticesofan$n$-dimensionalnondegeneratesimplex$S\subset B_n$\mathrm{.}Theinterpolationprojector$P:C(B_n)\to \Pi_1({\mathbb R}^n)$correspondingto$S$isdefinedbytheequalities$Pf\left(x^{(j)}\right) = f\left(x^{(j)}\right).$Denoteby$|P|_{B_n}$thenormof$P$ as an operator from $C(B_n)$ onto $C(B_n)$. We describe the approach in which $\mathrm{\parallel }P{\mathrm{\parallel }}_{B_n}$ can be estimated from below via the volume of $S$.