 Positive-product maximization on a convex set

Let $K\subseteq {\mathbb{R}}^2$ be compact and convex, and assume that it contains a point with both coordinates positive. The product $xy$ has a unique maximizer over $K\cap [0,\infty {)}^2$, and that maximizer lies in the positive quadrant. Compactness gives existence; the strictly positive attainable product excludes points on the axes. On the positive quadrant, maximizing $xy$ is equivalent to maximizing $\log x+\log y$, a strictly concave function. A segment between two distinct maximizers would contradict strict concavity, proving uniqueness.

The quadrant restriction matters. The line segment from $(-10,-10)$ to $(1,1)$ is a compact convex set meeting the positive quadrant, but its global product maximum over the entire segment is $100$ at the negative endpoint. The result concerns the positive feasible portion, unless the original feasible set is itself confined to the nonnegative quadrant.