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 $ξ(C;S)$ the minimal $τ>0$ such that $C$ is a subset of the simplex $τS$. By $α(C;S)$ we mean the minimal $τ>0$ such that $C$ is contained in a translate of $τS$. Earlier the author has proved the equalities $ξ(C;S)=(n+1)\max\limits_{1\leq j\leq n+1} \max\limits_{x\in C}(-λ_j(x))+1$ \ (if $C\not\subset S$), \ $α(C;S)= \sum\limits_{j=1}^{n+1} \max\limits_{x\in C} (-λ_j(x))+1.$ Here $λ_j$ 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 $B_n=\{x: \|x\|\leq 1\},$ where $\|x\|=\left(\sum\limits_{i=1}^n x_i^2 \right)^{1/2}.$ We establish various relations for $ξ(B_n;S)$ and $α(B_n;S)$, as well as we give their geometric interpretation.