= 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.
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