  An arrangement of k-semilines in the Euclidean (projective) plane or on the
2-sphere is called a k-fan if all semilines start from the same point. A k-fan
is an $\alpha$-partition for a probability measure $\mu$ if
$\mu(\sigma_i)=\alpha_i$ for each $i=1,...,k$ where $\{\sigma_i\}_{i=1}^k$ are
conical sectors associated with the k-fan and $\alpha = (\alpha_1,...
,\alpha_k)$. The set of all $\alpha = (\alpha_1,... ,\alpha_m)$ such that for
any collection of probability measures $\mu_1,... ,\mu_m$ there exists a common
$\alpha$-partition by a k-fan is denoted by ${\cal A}_{m,k}$. We prove, as a
central result of this paper, that ${\cal A}_{3,2} = \{(s,t)\in
\mathbb{R}^2\mid s+t=1 {\rm and} s,t>0\}$. The result follows from the fact
that under mild conditions there does not exist a $Q_{4n}$-equivariant map $f :
S^3\to V\setminus {\cal A}(\alpha)$ where ${\cal A}(\alpha)$ is a
$Q_{4n}$-invariant, linear subspace arrangement in a $Q_{4n}$-representation V,
where $Q_{4n}$ is the generalized quaternion group. This fact is established by
showing that an appropriate obstruction in the group $\Omega_1(Q_{4n})$ of
$Q_{4n}$-bordisms does not vanish.
