Let $n\ge2$ be an integer. Consider the hyperplane $H_n$ of ${\mathbb R}^n$ defined by the equation $x_1+\cdots+x_n=0$ and then the sector $P_n\subset H_n$ defined by the inequalities $x_1\le\cdots\le x_n$.
Given two points $\vec a,\vec b\in P_n$, the ordered spectra of all sums $A+B$, where $A$ (resp. $B$) is Hermitian and has spectrum $\vec a$ (resp. $\vec b$) form a convex polytope $Q(\vec a,\vec b)$ in $P_n$. This was conjectured by A. Horn in the 60's and proved by A. Knutson & T. Tao in the early 2000's.
The number of linear inequalities (the first ones are Weyl's inequalities, then Ky Fan, etc ...) which define $Q(\vec a,\vec b)$ is very large as $n$ increases. These polytopes are thus quite complicated when $n$ is not too small. But some of them could be way simpler, if a few inequalities turn out to imply the other ones.
Is there a tessalation of $P_n$ by polytopes of the form $Q(\vec a,\vec b)$ ? Presumably, periodic tiling is out of question.
