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Mathematics. Mechanics. Informatics

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Nevskii M. V., Ukhalov A. Y. Some Properties of 0/1-Simplices. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2018, vol. 18, iss. 3, pp. 305-315. DOI: 10.18500/1816-9791-2018-18-3-305-315

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Some Properties of 0/1-Simplices

Nevskii Mikhail V., P.G. Demidov Yaroslavl State University
Ukhalov Alexey Yu., P.G. Demidov Yaroslavl State University

Let n ∈ N, and let Q n = [0,1] n . For a nondegenerate simplex S ⊂ R n , by σS we mean the homothetic copy of S with center of homothety in the center of gravity of S and ratio of homothety σ. Put ξ(S) = min{σ > 1 : Q n ⊂ σS}, ξ n = min{ξ(S) : S ⊂ Q n }. By P we denote the interpolation projector from C(Q n ) onto the space of linear functions of n variables with the nodes in the vertices of a simplex S ⊂ Q n .LetkPkbethenormofP asanoperatorfromC(Q n )toC(Q n ),θ n = minkPk.Byξ ′ n andθ ′ n we denote the values analogous to ξ n and θ n , with the additional condition that corresponding simplices are 0/1-polytopes, i.e., their vertices coincide with vertices of Q n . In the present paper, we systematize general estimates of the numbers ξ ′ n , θ ′ n and also give their new estimates and precise values for some n. We prove that ξ ′n ≍ n, θ ′ n ≍ √ n. Let one vertex of 0/1-simplex S ∗ be an arbitrary vertex v of Q n and the other n vertice sare close to the vertex of the cube oppositetov. For 2 6 n 6 5, each simplex extremalin the sense of ξ ′ n coincides with S ∗ . The minimal n such that ξ(S ∗ ) > ξ ′ n is equal to 6. Denote by P ∗the interpolation projector with the nodes in the vertices of S ∗ . The minimal n such that kP ∗ k > θ ′ n is equal to 5.

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