# Balancing vectors and convex bodies

Studia Mathematica (1993)

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

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topBanaszczyk, 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 -

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