\noindent The simultaneous partition problems are classical problems of the combinatorial geometry which have the natural flavor of the equivariant topology. The $k$-fan partition problems have attracted a lot of attention \cite{ Aki2000}, \cite{ BaMa2001}, \cite{ BaMa2002} and forced some hard concrete combinatorial calculations in the equivariant cohomology \cite% { Bl-Vr-Ziv}. These problems can be reduced, by a beautiful scheme of \cite% { BaMa2001}, to a \textquotedblright typical\textquotedblright question of the existence of a $\mathbb{ D}_{ 2n}$ equivariant map $f:V_{ 2}(\mathbb{ R}% ^{ 3})\to W_{ n}-\cup \mathcal{ A}(\alpha)$, where $V_{ 2}(\mathbb{ R}% ^{ 3})$ is the space of all orthonormal 2-frames in $\mathbb{ R}^{ 3}$ and $% W_{ n}-\cup \mathcal{ A}(\alpha)$ is the complement of the appropriate arrangement. We introduce the \textit{ target extension scheme} which allow us to use the equivariant obstruction theory as a tool for proving that: for every two proper measures on the sphere $S^{ 2}$, and any $\alpha =(a,a+b,b)\in \mathbb{ R}_{ >0}^{ 3}$, there exists an $\alpha $-partition of theses measures by a 3-fan. \noindent The significance of these results, among other, is that, beside negative results \cite{ Bl-Vr-Ziv}, the equivariant obstruction theory can pull off some positive results, which were not attained by other means.
Categories: math.CO math.AT