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Choose a point of $K$ with both coordinates positive, so its product is positive. The set $K\cap [0,\infty {)}^2$ is nonempty and compact; the continuous function $xy$ therefore attains its maximum there. A maximizer cannot lie on either axis, where the product is zero. It consequently belongs to $K\cap P^{+}$ and also maximizes the product on that set.

For uniqueness, the function $\Phi (x,y)=\log x+\log y$ is strictly concave on the positive quadrant: its Hessian is $\mathrm{diag}(-1/x^2,-1/y^2)$, negative definite. If two distinct positive points attained the same maximal product, their midpoint would belong to $K\cap P^{+}$ by convexity and strict concavity would give a larger value of $\Phi$, hence a larger product. This contradiction proves existence and uniqueness of the positive product maximizer. This argument is positive-product maximization on a convex set; compactness should be applied to the closed quadrant intersection, not directly to the potentially noncompact set $K\cap P^{+}$.