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Can anyone give me an example of a function $f: \mathbb{R}^{n\times n} \rightarrow \mathbb{R}$, which is separately convex but not rank-one convex? By 'separately convex' I mean convexity in each matrix entry. By 'rank-one convex' I mean convexity in any rank-one direction. I would be totally happy with an example for $n=2$.

Many thanks in advance!

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up vote 1 down vote accepted

The function $$\begin{pmatrix}a & b \\ c & d \end{pmatrix}\mapsto ab$$ is linear in each entry, but not convex in the direction of the matrix $$\begin{pmatrix}1 & -1 \\ 0 & 0 \end{pmatrix}$$

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Thanks a lot for this example. –  Henry Wegener May 27 '12 at 18:54
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