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Let $C$ be a cone in $\mathbb{R}^d$, and let $x_1, \dots, x_k$ be its extreme rays. Suppose that the $x_i$ satisfy:

  • For all $i, j$, $\langle x_i, x_j \rangle \ge 0$,
  • There is a partition $A \cup B = \{x_1, \dots, x_k\}$ such that for every facet $F$ of $C$ and for every $x \in A \cap F, y \in B \cap F$ we have $\langle x,y\rangle = 0$.

Does it follow that every $x \in A$ is orthogonal to every $y \in B$?

I don't have any particularly strong reason to believe that this is true in general, but:

  • it's true in $\mathbb{R}^3$
  • it's true in $\mathbb{R}^d$ if the cone is full-dimensional and has at most $d+2$ extreme rays
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