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

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 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 $$C = {{Q}_{n}}$$.