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I'm wondering if there is any other characterization of the set of multivariate functions $f(x_1, ..., x_n) > 0$ such that, for any $i,$ $f$ is convex as a function of $x_i$ when $\{x_j\}_{j\neq i}$ are fixed at any values. Convex functions satisfy this condition, but it's a strictly weaker condition than convexity (see: Proof that a coordinate-wise convex function is convex?)? Products $\prod_{i=1}^n f_i(x_i)$ where each $f_i$ is convex and positive are in this set. Is this all? If there's some characterization without positivity constraints, I'd also be interested.

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