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Suppose $\Omega$ is a closed bounded subset of ${\mathbb R}^n,$ $S$ is an$n$-dimensional non-degenerate simplex, $\xi(\Omega;S):=\min \left\{\sigma\geq1: \, \Omega\subset \sigma S\right\}$. Here $\sigma S$ is the result ofhomothety of $S$ with respect to the center of gravity with coefficient$\sigma$. Let $d\geq n+1,$ $\varphi_1(x),\ldots,\varphi_d(x)$ be linearlyindependent monomials in $n$ variables, $\varphi_1(x)\equiv 1,$$\varphi_2(x)=x_1,\ \ldots, \ \varphi_{n+1}(x)=x_n.$ Put $\Pi:={\rmlin}(\varphi_1,\ldots,\varphi_d).$ The interpolation projector $P: C(\Omega)\to\Pi$ with a set of nodes $x^{(1)},\ldots, x^{(d)}$ $ \in \Omega$ is defined byequalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right).$ 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 theform $T(x):=(\varphi_2(x),\ldots,\varphi_d(x)). $ We have the followinginequalities: $\frac{1}{2}\left(1+\frac{1}{d-1}\right)\left(\|P\|_{\Omega}-1\right)+1$ $ \leq\xi(T(\Omega);S)\leq \frac{d}{2}\left(\|P\|_{\Omega}-1\right)+1. $ Here $S$ isthe $(d-1)$-dimensional simplex with vertices $T\left(x^{(j)}\right).$ Wediscuss this and other relations for polynomial interpolation of functionscontinuous on a segment. The results of numerical analysis are presented.

On a Geometric Approach to the Estimation of Interpolation Projectors