# On Numerical Characteristics of a Simplex and Their Estimates

## Abstract

Let *n* ∈N, and *Q*_{ n } = [0,1]^{ n } let be the *n*-dimensional unit cube. For a nondegenerate simplex *S* ⊂ R^{ n }, 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 } ⊂ σ*S*. By α(*S*) we denote the minimal σ > 0 such that *Q*_{ n } is contained in a translate of a simplex σ*S*. By *d*_{ i }(*S*) we mean the *i*th axial diameter of *S*, i. e., the maximum length of a segment contained in *S* and parallel to the *i*th 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 **ϰ**_{2} = \(\frac{{5 + 2\sqrt 5 }}{3}\) = 3.1573... Also we prove the new estimate ξ_{4} ≤ \(\frac{{19 + 5\sqrt {13} }}{9}\) = 4.1141..., which improves the earlier result ξ_{4} ≤ \(\frac{{13}}{3}\) = 4.33... Our conjecture is that ξ_{4} is precisely \(\frac{{19 + 5\sqrt {13} }}{9}\). Applying this value in numerical computations we achive the value **ϰ**_{4} = \(\frac{{4 + \sqrt {13} }}{5}\) =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 θ_{ 4 } =\(\frac{7}{3}\). Hence, the minimal *n* such that the above inequality has a strong form is equal to 4.

## Keywords

simplex cube coefficient of homothety axial diameter linear interpolation projector norm numerical methods## Preview

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