On one problem for a simplex and a cube in ℝn

Assume that S is a nondegenerate simplex in ℝ n . Denote by α( S ) the minimum σ > 0 such that the unit cube Q n := [0, 1] n belongs to a translate of σ S . In the case of σ( S ) ≠ 1, a translate of α( S ) S containing Q n is the image of S under homothety with the center at some point x ∈ ℝ n . In this paper, we present the following computational formula for x . Denote by x ( j ) ( j = 1, ..., n + 1) the vertices of S . Let A be an n + 1-by- n + 1 matrix such that its rows contain the coordinates of x ( j ) ; here, the last column of A consists of ones. Assume that A −1 = ( l ij ). Then, the coordinates of x are the numbers Since α( S ) ≠ 1, the denominator in the right-hand side of this equality is distinct from zero. In addition, we present estimates for norms of projections under the linear interpolation of continuous functions given on Q n .