Positive-product maximization on a convex set

A compact convex subset of ${\mathbb{R}}^2$ meeting the positive quadrant has a unique positive point maximizing $xy$. Existence follows by maximizing the continuous product on the intersection with the closed nonnegative quadrant; a positive attainable value excludes its axes. Uniqueness follows from strict concavity of $\log x+\log y$ on the positive quadrant. This is a useful way to optimize a product even though the product itself is not concave.