Write

$$P^{+}=\left\{(x, y) \in \mathbb{R}^{2}: x, y>0\right\} .$$

Suppose that $K$ is a convex, compact subset of $\mathbb{R}^{2}$ with $K \cap P^{+} \neq \emptyset$. Show that there is a unique point $\left(x_{0}, y_{0}\right) \in K \cap P^{+}$such that

$$x y \leqslant x_{0} y_{0}$$

for all $(x, y) \in K \cap P^{+}$.