On the Minimal Positive Homothetic Image of a Simplex Containing a Convex Body

M. V. Nevskii

Matematičeskie zametki, Tome 93 (2013) no. 3, pp. 448-456

Résumé

Let $C$ be a convex body, and let $S$ be a nondegenerate simplex in $\mathbb R^n$. It is proved that the minimal coefficient $\sigma>0$ for which the translate of $\sigma S$ contains $C$ is $$ \sum_{j=1}^{n+1}\max_{x\in C}(-\lambda_j(x))+1, $$ where $\lambda_1(x),\dots,\lambda_{n+1}(x)$ are the barycentric coordinates of the point $x\in\mathbb R^n$ with respect to $S$. In the case $C=[0,1]^n$, this quantity is reduced to the form $\sum_{i=1}^n 1/d_i(S)$, where $d_i(S)$ is the $i$th axial diameter of $S$, i.e., the maximal length of the segment from $S$ parallel to the $i$th coordinate axis.

Détail

Keywords: $n$-dimensional simplex, homothetic image of a simplex, translate, axial diameter of a simplex, barycentric coordinates, convex body.