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Product maximizer on a compact convex subset of the positive orthant

On the positive orthant,

F (x) = \sum_{j = 1}^{n} lo g x_{j}

is strictly concave. Its unique maximizer

x^{*}

on a compact convex feasible set satisfies the first-order inequality

\nabla F (x^{*}) \cdot (x - x^{*}) \leq 0,

equivalently

\sum_{j = 1}^{n} \frac{x _{j}}{x _{j}^{*}} \leq n .

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