Suppose $f(x_1, \dots, x_n): \mathbb{R}^n \to \mathbb{R}$. Is there an example of such $f$ that is convex in every argument and not convex? Are there any additional conditions, under which 'marginal' convexity would be sufficient to call $f$ convex?
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5Duplicate of Proof that a coordinate-wise convex function is convex?. See also Convexity definition confusion. – dxiv May 15 '17 at 00:13
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Convexity along the axes is too little for the function for the overall convexity. To be convex the function must be convex not only along the axes, but along all lines in the domain. – A.Γ. May 15 '17 at 00:31