I am trying to prove the difference of two algorithms' asymptotic mean square error. Eventually, I get an expression like $x^TAy$, where $A$ is a positive definite matrix. I would love to see under which conditions one algorithm can "beat the other", i.e., I am looking for the $\mathcal D \subset \Bbb R^p \times \Bbb R^p$ such that
$$\left( \forall (x,y) \in \mathcal D \right) \left( x^T A y \ge 0 \right)$$
where $x, y \in \Bbb R^p$ and $p \gg 2$. For example, if we treat it as a constrained optimization problem
$$S := \min x^TAy$$
subject to "some constraints", then $S=0$. What constraints can I put here? Any ideas are appreciated.