On Some Problems Related to a Simplex and a Ball

Abstract

Let $C$ be a convex body and let $S$ be a nondegenerate simplex in ${\mathbb{R}}^n$. Denote by $\xi (C;S)$ the minimal $\tau >0$ such that $C$ is a subset of the simplex $\tau S$. By $\alpha (C;S)$ we mean the minimal $\tau >0$ such that $C$ is contained in a translate of $\tau S$. Earlier the author has proved the equalities $\xi (C;S)=(n+1)\underset{1\le j\le n+1}{\max }\underset{x\in C}{\max }(-{\lambda }_j(x))+1$ \ (if $C\not\subset S$), \ $\alpha (C;S)=\sum _{j=1}^{n+1}\underset{x\in C}{\max }(-{\lambda }_j(x))+1.$ Here ${\lambda }_j$ are linear functions called the basic Lagrange polynomials corresponding to $S$. In his previous papers, the author has investigated these formulae if $C=[0,1{]}^n$. The present paper is related to the case when $C$ coincides with the unit Euclidean ball $B_n=\{x:\Vert x\Vert \le 1\},$ where $\Vert x\Vert ={\left(\sum _{i=1}^n{x}_i^2\right)}^{1/2}.$ We establish various relations for $\xi (B_n;S)$ and $\alpha (B_n;S)$, as well as we give their geometric interpretation.

13 pages