  This paper contains a survey of results obtained by the authors mostly during
the past few years and published by 2021. In particular, we present the best of
known estimates of numerical characteristics related to the research theme.
  Sections: 1. Introduction. 2. The case when $n+1$ is an Hadamard number. 3.
Estimates for the minimal absorption index of a cube by a simplex. 4. Estimates
for the minimal norm of a projector in linear interpolation on a cube in
${\mathbb R}^n$. 5. Estimates of numbers $\xi_n^\prime$ and $\theta_n^\prime$.
6. Simplices satisfying the inclusions $S\subset Q_n\subset nS$. 7. Perfect
simplices. 8. Equisecting simplices. 9. Properties of $(0,1)$-matrices of order
$n$ having maximal determinant. 10. Problems for a simplex and a Euclidean
ball. 11. Linear interpolation on a Euclidean ball.
  Bibliography: 56 titles. Keywords: simplex, cube, Euclidean ball, homothety,
axial diameter, absorption index, Hadamard number, interpolation, projector,
norm, estimate.
