  Let $H$ be a Hilbert space. Using Ball's solution of the "complex plank
problem" we prove that the following properties of a sequence $a_n>0$ are
equivalent: (1) There is a sequence $x_n \in H$ with $\|x_n\|=a_n$, having 0 as
a weak cluster point; (2) $\sum_1^\infty a_n^{-2}=\infty$. Using this result we
show that a natural idea of generalization of Ball's "complex plank" result to
cylinders with $k$-dimensional base fails already for $k=3$. We discuss also
generalizations of "weak cluster points" result to other Banach spaces and
relations with cotype.
