Balancing vectors and convex bodies

Let U, V be two symmetric convex bodies in ${\displaystyle {ℝ}^n}$ and |U|, |V| their n-dimensional volumes. It is proved that there exist vectors ${\displaystyle u_1,...,u_n\in U}$ such that, for each choice of signs ${\displaystyle {\epsilon }_1,...,{\epsilon }_n=\pm 1}$, one has ${\displaystyle {\epsilon }_1{u}_1+...+{\epsilon }_n{u}_n\notin rV}$ where ${\displaystyle r={(2\pi e^2)}^{-\frac{1}{2}}{n}^{\frac{1}{2}}{\left(\frac{\left|U\right|}{\left|V\right|}\right)}^{\frac{1}{n}}}$. Hence it is deduced that if a metrizable locally convex space is not nuclear, then it contains a null sequence ${\displaystyle \left(u_n\right)}$ such that the series ${\displaystyle {\sum }_{n=1}^{\infty }{\epsilon }_n{u}_{\pi \left(n\right)}}$ is divergent for any choice of signs ${\displaystyle {\epsilon }_n=\pm 1}$ and any permutation π of indices.