Balancing vectors and convex bodies

Banaszczyk, Wojciech

Studia Mathematica, Tome 104 (1993), p. 93-100 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $ε_{1}, . . ., ε_{n} = \pm 1$ , one has $ε_{1} u_{1} + . . . + ε_{n} u_{n} \notin r V$ where $r = {(2 π e^{2})}^{- 1 / 2} n^{1 / 2} (| U | / {| V |)}^{1 / n}$ . 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 $\sum_{n = 1}^{\infty} ε_{n} u_{π (n)}$ is divergent for any choice of signs $ε_{n} = \pm 1$ and any permutation π of indices.

Publié le : 1993-01-01

Zbl 0810.46009

EUDML-ID : urn:eudml:doc:216005

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     author = {Wojciech Banaszczyk},
     title = {Balancing vectors and convex bodies},
     journal = {Studia Mathematica},
     volume = {104},
     year = {1993},
     pages = {93-100},
     zbl = {0810.46009},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv106i1p93bwm}
}

Banaszczyk, Wojciech. Balancing vectors and convex bodies. Studia Mathematica, Tome 104 (1993) pp. 93-100. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv106i1p93bwm/

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