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Mat. Zametki, 2013, Volume 93, Issue 3, Pages 448–456 (Mi mz9200)  

This article is cited in 1 scientific paper (total in 1 paper)

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

M. V. Nevskii

P. G. Demidov Yaroslavl State University

Abstract: 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 $\lambda_1(x),…,\lambda_{n+1}(x)$ 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.

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

DOI: https://doi.org/10.4213/mzm9200

Full text: PDF file (490 kB)
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English version:
Mathematical Notes, 2013, 93:3, 470–478

Bibliographic databases:

Document Type: Article
UDC: 514.17
Received: 05.07.2011
Revised: 14.02.2012

Citation: M. V. Nevskii, “On the Minimal Positive Homothetic Image of a Simplex Containing a Convex Body”, Mat. Zametki, 93:3 (2013), 448–456; Math. Notes, 93:3 (2013), 470–478

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. M. V. Nevskii, “Ob odnoi zadache dlya simpleksa i kuba v ${\mathbb R}^n$”, Model. i analiz inform. sistem, 20:3 (2013), 77–85  mathnet
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