On Numerical Characteristics of a Simplex and Their Estimates

Let n ∈N, and Q_n = [0,1]ⁿ let be the n-dimensional unit cube. For a nondegenerate simplex S ⊂ Rⁿ, by σ^S we denote the homothetic image of with center of homothety in the center of gravity of S and ratio of homothety σ. We apply the following numerical characteristics of a simplex. Denote by ξ(S) the minimal σ > 0 with the property Q_n ⊂ σS. By α(S) we denote the minimal σ > 0 such that Q_n is contained in a translate of a simplex σS. By d_i(S) we mean the ith axial diameter of S, i. e., the maximum length of a segment contained in S and parallel to the ith coordinate axis. We apply the computational formulae for ξ(S), α(S), d_i(S) which have been proved by the first author. In the paper we discuss the case S ⊂ Q_n. Let ξ_n = min{ξ(S): S ⊂ Q_n}. Earlier the first author formulated the conjecture: if ξ(S) = ξ_n, then α(S) = ξ(S). He proved this statement for n = 2 and the case when n + 1 is an Hadamard number, i. e., there exist an Hadamard matrix of order n + 1. The following conjecture is more strong proposition: for each n, there exist γ ≥ 1, not depending on S ⊂ Q_n, such that ξ(S) − α(S) ≤ γ(ξ(S) − ξ_n). By ϰ_n we denote the minimal γ with such a property. If n + 1 is an Hadamard number, then the precise value of ϰ_n is 1. The existence of ϰ_n for other n was unclear. In this paper with the use of computer methods we obtain an equality ϰ₂ = \(\frac{{5 + 2\sqrt 5 }}{3}\) = 3.1573... Also we prove the new estimate ξ₄ ≤ \(\frac{{19 + 5\sqrt {13} }}{9}\) = 4.1141..., which improves the earlier result ξ₄ ≤ \(\frac{{13}}{3}\) = 4.33... Our conjecture is that ξ₄ is precisely \(\frac{{19 + 5\sqrt {13} }}{9}\). Applying this value in numerical computations we achive the value ϰ₄ = \(\frac{{4 + \sqrt {13} }}{5}\) =1.5211... Denote by θ_n the minimal norm of interpolation projector onto the space of linear functions of n variables as an operator from C(Q_n) in C(Q_n). It is known that, for each n, ξ_n ≤ \(\frac{{n + 1}}{2}({\theta _n} - 1) + 1\), and for n = 1, 2,3, 7 here we have an equality. Using computer methods we obtain the result θ₄ =\(\frac{7}{3}\). Hence, the minimal n such that the above inequality has a strong form is equal to 4.