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Standard facts about separating linear functionals will be used to determine how two cones $C$ and $D$ and their duals $C^*$ and $D^*$ may overlap. When $T\:V\rightarrow W$ is linear and $K \subset V$ and $D\subset W$ are cones, these results will be applied to $C=T(K)$ and $D$, giving a unified treatment of several theorems of the alternate which explain when $C$ contains an interior point of $D$. The case when $V=W$ is the space $H$ of $n\times n$ Hermitian matrices, $D$ is the $n\times n$ positive semidefinite matrices, and $T(X) = AX + X^*A$ yields new and known results about the existence of block diagonal $X$’s satisfying the Lyapunov condition: $T(X)$ is an interior point of $D$. For the same $V$, $W$ and $D$, $ T(X)=X-B^*XB$ will be studied for certain cones $K$ of entry-wise nonnegative $X$’s.
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