Majorization of $(5,5,0) $ by $(10,0,0)$

Question

Majorization definition: $x $ is majorized by $y$ iff $x$ is in the convex hull of all the points formed by permuting the components of $y$.

Question: $(5,5,0) $ is majorized by $(10,0,0)$ according to the alternate definition of majorization given at wikipedia.

My confusion is: the convex hull formed by $(10,0,0)$ is the planar equilateral-triangle with three vertices at $(10,0,0), (0,10,0)$ and $ (0,0,10)$; and $(5,5,0)$ is not the part of this convex set. I see the two definitions contradicting.

Answer 1

We have $(5,5,0) = 0.5(10,0,0) + 0.5(0,10,0)+0(0,0,10)$, so it is in the convex hull.