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- W3100561307 abstract "Let (C) be a convex body and let (S) be a nondegenerate simplex in ({mathbb R}^n). Denote by (tau S) the image of (S) under homothety with a center of homothety in the center of gravity of (S) and the ratio (tau). We mean by (xi(C;S)) the minimal (tau>0) such that (C) is a subset of the simplex (tau S). Define (alpha(C;S)) as the minimal (tau>0) such that (C) is contained in a translate of (tau S). Earlier the author has proved the equalities (xi(C;S)=(n+1)maxlimits_{1leq jleq n+1}maxlimits_{xin C}(-lambda_j(x))+1) (if (Cnotsubset S)), (alpha(C;S)=sumlimits_{j=1}^{n+1} maxlimits_{xin C} (-lambda_j(x))+1.)Here (lambda_j) are the linear functions that are called the basic Lagrange polynomials corresponding to (S). The numbers (lambda_j(x),ldots, lambda_{n+1}(x)) are the barycentric coordinates of a point (xin{mathbb R}^n). In his previous papers, the author investigated these formulae in the case when (C) is the (n)-dimensional unit cube (Q_n=[0,1]^n). The present paper is related to the case when (C) coincides with the unit Euclidean ball (B_n={x: |x|leq 1},) where (|x|=left(sumlimits_{i=1}^n x_i^2 right)^{1/2}.) We establish various relations for (xi(B_n;S)) and (alpha(B_n;S)), as well as we give their geometric interpretation. For example, if (lambda_j(x)=l_{1j}x_1+ldots+l_{nj}x_n+l_{n+1,j},) then (alpha(B_n;S)=sumlimits_{j=1}^{n+1}left(sumlimits_{i=1}^n l_{ij}^2right)^{1/2}). The minimal possible value of each characteristics (xi(B_n;S)) and (alpha(B_n;S)) for (Ssubset B_n) is equal to (n). This value corresponds to a regular simplex inscribed into (B_n). Also we compare our results with those obtained in the case (C=Q_n)." @default.
- W3100561307 created "2020-11-23" @default.
- W3100561307 creator A5001284414 @default.
- W3100561307 date "2018-12-19" @default.
- W3100561307 modified "2023-10-03" @default.
- W3100561307 title "On Some Problems for a Simplex and a Ball in R<sup>n</sup>" @default.
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- W3100561307 doi "https://doi.org/10.18255/1818-1015-2018-6-680-691" @default.
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