the point $ \bP^*_\mu $, which is defined in Theorem \[YorTh\], can be considered to be the [*center*]{} of the set of solutions $ \sP_\mu $.

\[center point\] Let $ \mu $ be given. Then, for any $ \bP \in \sP_\mu $, it holds that $$\bP^*_\mu = \int_G T_g(\bP) \nu(dg) . 
\label{center point 1}$$

For any bounded measurable function $ \phi $ on $ G $, we have $$\begin{aligned}
&& \int_G \phi(h) \int_G T_g(\bP)(\eta_k \in dh) \nu(dg) 
\\
&& = \int_G \bE {\left[  \phi(g \cdot \eta_k)  \right]} \nu(dg) = \int_G \phi(g) \nu(dg) \end{aligned}$$ for any $ k \in -\N $. This implies that all marginal distributions of the RHS of are uniform on $ G $.

Now we are in a position to give the precise statement of Theorem \[Intro action\].

\[Gal1\] Let $ \mu $ be given. Then the action of $ (T_g: \, g \in G) $ restricted on $ \ext $ is transitive: That is, if $ \bP^1 $ and $ \bP^2 $ are two solutions in $ \ext $, then there exists an element $ g \in G $ such that $ \bP^1 = T_g (\bP^2) $.

Let an extremal point $ \bP^o \in \ext $ 