Geometric Estimates in Linear Interpolation on a Cube and a Ball

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.