  The paper contains a survey of the results obtained by the author in recent
years. These results concern the application in multivariate polynomial
interpolation of some geometric constructions and methods. In particular, we
give estimates of the projector's norms through the characteristics of sets
associated with homothety. The known exact values and nowaday best estimates of
these norms are given. Also we formulate some open problems. The survey is
dedicated to Professor Yuri Brudnyi in connection with the upcoming 90th
anniversary of his birth. Contents: Introduction. 1. Notation and
preliminaries. 2. The values of $\alpha(Q_n;S)$ and $\xi(Q_n;S)$. 3. The values
of $\alpha(B_n;S)$ and $\xi(B_n;S)$. 4. Estimates for $\theta_n(Q_n)$. 5.
Legendre polynomials and the measure of $E_{n,\gamma}$. 6. Inequalities
$\theta_n(Q_n)>c\sqrt{n}$ and $\theta_n(B_n)>c\sqrt{n}$. 7. The norm
$\|P\|_{B}$ for an inscribed regular simplex. 8. A theorem on a simplex and its
minimal ellipsoid. 9. The value of $\theta_n(B_n)$. 10. Interpolation by wider
polynomial spaces.
