On a geometric approach to the estimation of interpolation projectors

Suppose $\Omega$ is a closed bounded subset of ${\mathbb R}^n,$ $S$ is an $n$-dimensional non-degenerate simplex, $\xi(\Omega;S):=$ min {$\sigma\geqslant 1: \Omega\subset \sigma S$}. Here $\sigma S$ is the result of homothety of $S$ with respect to the center of gravity with coefficient $\sigma$. Let $d\geqslant n+1,$ $\varphi_1(x),\ldots,\varphi_d(x)$ be linearly independent monomials in $n$ variables, and $\varphi_1(x)\equiv 1,$ $\varphi_2(x)=x_1,\ \ldots, \varphi_{n+1}(x)=x_n.$ Put $\Pi:=$lin$(\varphi_1,\ldots,\varphi_d).$ The interpolation projector $P: C(\Omega)\to \Pi$ with a set of nodes $x^{(1)},\ldots, x^{(d)} \in \Omega$ is defined by equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right).$ Denote by $\|P\|_{\Omega}$ the norm of $P$ as an operator from $C(\Omega)$ to $C(\Omega)$ . Consider the mapping $T:{\mathbb R}^n\to {\mathbb R}^{d-1}$ of the form $T(x):=(\varphi_2(x),\ldots,\varphi_d(x)). $ We have $ \frac{1}{2}\left(1+\frac{1}{d-1}\right)\left(\|P\|_{\Omega}-1\right)+1 \leqslant \xi(T(\Omega);S)\leqslant \frac{d}{2}\left(\|P\|_{\Omega}-1\right)+1, $ where $S$ is a $(d-1)$-dimensional simplex with vertices $T\left(x^{(j)}\right).$ We discuss this and other relations for polynomial interpolation of functions continuous on a segment. Some results of numerical analysis are presented.