# Balancing vectors and convex bodies

Studia Mathematica (1993)

• Volume: 106, Issue: 1, page 93-100
• ISSN: 0039-3223

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## Abstract

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Let U, V be two symmetric convex bodies in ${ℝ}^{n}$ and |U|, |V| their n-dimensional volumes. It is proved that there exist vectors ${u}_{1},...,{u}_{n}\in U$ such that, for each choice of signs ${\epsilon }_{1},...,{\epsilon }_{n}=±1$, one has ${\epsilon }_{1}{u}_{1}+...+{\epsilon }_{n}{u}_{n}\notin rV$ where $r={\left(2\pi {e}^{2}\right)}^{-1/2}{n}^{1/2}\left(|U|/{|V|\right)}^{1/n}$. Hence it is deduced that if a metrizable locally convex space is not nuclear, then it contains a null sequence $\left({u}_{n}\right)$ such that the series ${\sum }_{n=1}^{\infty }{\epsilon }_{n}{u}_{\pi \left(n\right)}$ is divergent for any choice of signs ${\epsilon }_{n}=±1$ and any permutation π of indices.

## How to cite

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Banaszczyk, Wojciech. "Balancing vectors and convex bodies." Studia Mathematica 106.1 (1993): 93-100. <http://eudml.org/doc/216005>.

@article{Banaszczyk1993,
abstract = {Let U, V be two symmetric convex bodies in $ℝ^n$ and |U|, |V| their n-dimensional volumes. It is proved that there exist vectors $u_1,...,u_n ∈ U$ such that, for each choice of signs $ε_1,...,ε_n = ± 1$, one has $ε_1 u_1 + ... + ε_n u_n ∉ rV$ where $r = (2πe^2)^\{-1/2\} n^\{1/2\}(|U|/|V|)^\{1/n\}$. Hence it is deduced that if a metrizable locally convex space is not nuclear, then it contains a null sequence $(u_n)$ such that the series $∑_\{n = 1\}^∞ ε_n u_\{π(n)\}$ is divergent for any choice of signs $ε_n = ± 1$ and any permutation π of indices.},
author = {Banaszczyk, Wojciech},
journal = {Studia Mathematica},
keywords = {balancing vectors; Steinitz constant; symmetric convex bodies; metrizable locally convex space; nuclear},
language = {eng},
number = {1},
pages = {93-100},
title = {Balancing vectors and convex bodies},
url = {http://eudml.org/doc/216005},
volume = {106},
year = {1993},
}

TY - JOUR
AU - Banaszczyk, Wojciech
TI - Balancing vectors and convex bodies
JO - Studia Mathematica
PY - 1993
VL - 106
IS - 1
SP - 93
EP - 100
AB - Let U, V be two symmetric convex bodies in $ℝ^n$ and |U|, |V| their n-dimensional volumes. It is proved that there exist vectors $u_1,...,u_n ∈ U$ such that, for each choice of signs $ε_1,...,ε_n = ± 1$, one has $ε_1 u_1 + ... + ε_n u_n ∉ rV$ where $r = (2πe^2)^{-1/2} n^{1/2}(|U|/|V|)^{1/n}$. Hence it is deduced that if a metrizable locally convex space is not nuclear, then it contains a null sequence $(u_n)$ such that the series $∑_{n = 1}^∞ ε_n u_{π(n)}$ is divergent for any choice of signs $ε_n = ± 1$ and any permutation π of indices.
LA - eng
KW - balancing vectors; Steinitz constant; symmetric convex bodies; metrizable locally convex space; nuclear
UR - http://eudml.org/doc/216005
ER -

## References

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