# New Estimates of Numerical Values Related to a Simplex

## Abstract

Let *n* ∈N, and let *Q*_{ n } = [0 1]^{ n }. For a nondegenerate simplex *S* ⊂ R^{ n }, by σ*S* 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 } ⊂ σ*S*. By α(*S*) let us denote the minimal σ > 0 such that *Q*_{ n } is contained in a translate of σ*S*. By *d*_{ i }(*S*) we denote the *i*th axial diameter of *S*, i. e. the maximum length of the segment contained in *S* and parallel to the *i*th 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 **ϰ**_{1} = \(\frac{1}{2}\); 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 ξ_{ 5 } = *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 **ϰ**_{ 5 } = 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* ⊂ *Q*_{5} exists simplex such that the boundary of the simplex 5*S* contains all the vertices of the cube *Q*_{5}. We describe one-parameter family of simplices contained in *Q*_{5} 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 ξ_{6} < 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 }).

### Keywords

simplex cube homothety axial diameter interpolation projector numerical methods## Preview

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