  For a fixed unit vector $a=(a_1,a_2,\ldots,a_n)\in S^{n-1}$, we consider the
$2^n$ sign vectors
$\varepsilon=(\varepsilon^1,\varepsilon^2,\ldots,\varepsilon^n)\in \{+1,-1\}^n$
and the corresponding scalar products $\varepsilon\cdot a=\sum_{i=1}^n
\varepsilon^ia_i$. In this paper we will solve for $n=1,2,\ldots,9$ an old
conjecture stating that of the $2^n$ sums of the form $\sum\pm a_i$ it is
impossible that there are more with $|\sum_{i=1}^n \pm a_i|>1$ than there are
with $|\sum_{i=1}^n \pm a_i|\leq1$. Although the problem has been solved
completely in case the $a_i$'s are equal, the more general problem with
possible non-equal $a_i$'s remains open for values of $n\geq 10$. The present
method can also be used for $n\geq 10$, but unfortunately the technical
difficulties seem to grow exponentially with $n$ and no "induction type of
argument" has been found. The conjecture has an appealing reformulation in
probability theory and in geometry. In probability theory the results lead to
upper bounds which are much better than for instance Chebyshevnequalities.
