Automatic Control and Computer Sciences

, Volume 51, Issue 7, pp 770–782

New Estimates of Numerical Values Related to a Simplex

Article

Abstract

Let n ∈N, and let Q n = [0 1] n . For a nondegenerate simplex S ⊂ R n , 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): SQ 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 SQ 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 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 SQ5 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 ).

Keywords

simplex cube homothety axial diameter interpolation projector numerical methods

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