On Numerical Characteristics of a Simplex and Their Estimates

Let n ∈N, and Qn = [0,1]n let be the n-dimensional unit cube. For a nondegenerate simplex S ⊂ Rn, by σS 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 Qn ⊂ σS. By α(S) we denote the minimal σ > 0 such that Qn is contained in a translate of a simplex σS. By di(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), di(S) which have been proved by the first author. In the paper we discuss the case S ⊂ Qn. Let ξn = min{ξ(S): S ⊂ Qn}. Earlier the first author formulated the conjecture: if ξ(S) = ξn, 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 ⊂ Qn, such that ξ(S) − α(S) ≤S) − ξn). By ϰn we denote the minimal γ with such a property. If n + 1 is an Hadamard number, then the precise value of ϰn is 1. The existence of ϰn for other n was unclear. In this paper with the use of computer methods we obtain an equality ϰ2 = 5+253 = 3.1573.. Also we prove the new estimate ξ4 ≤ 19+5139 = 4.1141.., which improves the earlier result ξ4 ≤ 133 = 4.33.. Our conjecture is that ξ4 is precisely 19+5139. Applying this value in numerical computations we achive the value ϰ4 = 4+135 =1.5211.. Denote by θn the minimal norm of interpolation projector onto the space of linear functions of n variables as an operator from C(Qn) in C(Qn). It is known that, for each n, ξn ≤ n+12(θn−1)+1, and for n = 1, 2,3, 7 here we have an equality. Using computer methods we obtain the result θ4 =73. Hence, the minimal n such that the above inequality has a strong form is equal to 4. © 2017, Allerton Press, Inc.