Why is there this contradiction or what is wrong In the second paragraph on Page 30 of this published paper, it says that the intersection of the convex
hull of points $(\alpha_{1}+\beta_{P_{1}},\alpha_{2}+\beta_{P_{2}})$
with the convex hull of points $(\beta_{1}+\alpha_{P_{1}},\beta_{2}+\alpha_{P_{2}})$
contains the point $(\gamma_{1},\gamma_{2})$. For example, if
$\alpha_{1}=2$, $\alpha_{2}=0$, $\beta_{1}=3$, and $\beta_{2}=2$,
then the intersection is a line segment between $(3,4)$ and $(4,3)$
. But don't the eigenvalues of the the sum matrix actually lie on
the line segment between $(1,5)$and $(3,4)$? or what is wrong?
 A: This appears to be a simple case of misunderstanding sloppy notation. I think what you were going for is to fix $P$ to be the non-identity permutation. But what the author appears to be doing is different: e allows $P$ to range over all permutations, and taking the convex hulls among each "type" of points.
Doing this, you get that the $\gamma$-point is in the intersection of


*

*the convex hull of $\{(\alpha_{1}+\beta_{1},\alpha_{2}+\beta_{2}), (\alpha_{1}+\beta_{2},\alpha_{2}+\beta_{1})\}$, i.e. the straight line with endpoints $(5,2)$ and $(4,3)$, and

*the convex hull of $\{(\beta_{1}+\alpha_{1},\beta_{2}+\alpha_{2}),(\beta_{1}+\alpha_{2},\beta_{2}+\alpha_{1})\}$, i.e. the straight line with endpoints $(5,2)$ and $(3,4)$.


Or, in fewer words, it is in the line segment with endpoints $(5,2)$ and $(4,3)$.
This is not on the line segment between $(1,5)$ and $(3,4)$; I am not sure where you got these figures from, but by constructing the diagonal matrices from the alphas and betas, you will see that $(5,2)$, in particular, really must be in the acceptable region.
