Geodesic
Balancing vectors and convex bodies
Studia Mathematica, Tome 106 (1993) no. 1, pp. 93-100

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

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.
DOI : 10.4064/sm-106-1-93-100
Keywords: balancing vectors, Steinitz constant

Wojciech Banaszczyk 1

1 
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Wojciech Banaszczyk. Balancing vectors and convex bodies. Studia Mathematica, Tome 106 (1993) no. 1, pp. 93-100. doi: 10.4064/sm-106-1-93-100

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