New Estimates of Numerical Values Related to a Simplex

Let n ∈N, and let Q_n = [0 1]ⁿ. For a nondegenerate simplex S ⊂ Rⁿ, by σS we denote the homothetic copy of S with center of homothety in the center of gravity of S and ratio of homothety σ. By ξ(S) we mean the minimal σ > 0 such that Q_n ⊂ σS. By α(S) let us denote the minimal σ > 0 such that Q_n is contained in a translate of σS. By d_i(S) we denote the ith axial diameter of S, i. e. the maximum length of the segment contained in S and parallel to the ith coordinate axis. Formulae for ξ(S), α(S), d_i(S) were proved earlier by the first author. Define ξ_n = min{ξ(S): S ⊂ Q_n}. Always we have ξ_n ≥ n. We discuss some conjectures formulated in the previous papers. One of these conjectures is the following. For every n, there exists γ > 0, not depending on S ⊂ Q_n, such that an inequality ξ(S) − α(S) ≤ γ(ξ(S) − ξ_n) holds. Denote by ϰ_n the minimal γ with such a property. We prove that ϰ₁ = \(\frac{1}{2}\); for n > 1, we obtain ϰ_n ≥ 1. If n > 1 and ξ_n = n, then ϰ_n = 1. The equality holds ξ_n = n if n + 1is an Hadamard number, i. e. there exists an Hadamard matrix of order n + 1. This proposition is known; we give one more proof with the direct use of Hadamard matrices. We prove that ξ₅ = 5. Therefore, there exist n such that n + 1 is not an Hadamard number and nevertheless ξ_n = n. The minimal n with such a property is equal to 5. This involves ϰ₅ = 1 and also disproves the following previous conjecture of the first author concerning the characterization of Hadamard numbers in terms of homothety of simplices: n + 1 is an Hadamard number if and only if ξ_n = n. This statement is valid only in one direction. There S ⊂ Q₅ exists simplex such that the boundary of the simplex 5S contains all the vertices of the cube Q₅. We describe one-parameter family of simplices contained in Q₅ with the property α(S) = ξ(S) = 5. These simplices were found with the use of numerical and symbolic computations. Another new result is an inequality ξ₆ < 6.0166. We also systematize some of our estimates of numbers ξ_n, θ_n, ϰ_n provided by the present day. The symbol θ_n denotes the minimal norm of interpolation projector on the space of linear functions of n variables as an operator from C(Q_n) to C(Q_n).