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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?

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  • $\begingroup$ Any helpful answers would be highly appreciated. $\endgroup$ – taepia Aug 5 '13 at 7:05
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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.

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  • $\begingroup$ dear Eric, thank you for your answer, but actually the intersection of the line segment line with endpoints (5,2) and (4,3) and the line with endpoints (5,2) and (3,4) is the line segment line with endpoints (5,2) and (4,3), because they lie on the same straight line. So it should not be a single point. $\endgroup$ – taepia Aug 5 '13 at 16:32
  • $\begingroup$ @taepia: I figured it would be something silly like that… Thank you. $\endgroup$ – Eric Stucky Aug 5 '13 at 18:09

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