On Some Problems for a Simplex and a Ball in \({{\mathbb{R}}^{n}}\)

Resumo

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 the homothety with center of homothety in the center of gravity of \(S\) and ratio of homothety \(\tau \). 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 \(\xi (C;S) = (n + 1)\mathop {\max }\limits_{1 \leqslant j \leqslant n + 1} \mathop {\max }\limits_{x \in C} ( - {{\lambda }_{j}}(x)) + 1\) (if \(C { \text{⊄} }S\)), \(\alpha (C;S) = \sum\nolimits_{j = 1}^{n + 1} {\mathop {\max }\limits_{x \in C} ( - {{\lambda }_{j}}(x)) + 1.} \) Here \({{\lambda }_{j}}\) are linear functions 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 \(x \in {{\mathbb{R}}^{n}}\). In his previous papers, the author has 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 \({{B}_{n}} = \{ x:\left| {\left| x \right|} \right| \leqslant 1\} ,\) where \(\left| {\left| x \right|} \right| = \mathop {\left( {\sum\nolimits_{i = 1}^n {x_{i}^{2}} \,} \right)}\nolimits^{1/2} .\) 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 \(\alpha ({{B}_{n}};S) = \sum\nolimits_{j = 1}^{n + 1} {{{{\left( {\sum\nolimits_{i = 1}^n {l_{{ij}}^{2}} } \right)}}^{{{\text{1}}{\text{/}}{\text{2}}}}}} \). The minimal possible value of each characteristic \(\xi ({{B}_{n}};S)\) and \(\alpha ({{B}_{n}};S)\) for \(S \subset {{B}_{n}}\) 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 \(C = {{Q}_{n}}\).